Filed by Envision Healthcare Holdings, Inc.

Pursuant to Rule 425 under the Securities Act of 1933, as amended

and deemed filed pursuant to Rule 14a-12

under the Securities Exchange Act of 1934, as amended

Subject Company: Envision Healthcare Holdings, Inc.

Commission File No. 001-36048

[The following is an email sent to employees of AmSurg Corp. and Envision Healthcare Holdings, Inc. on October 27, 2016.]

As we continue to make progress toward closing the merger of Envision Healthcare and AMSURG, we want to introduce you to the combined entity’s future leaders and business lines. In this series, we profile Bob Coward, the current President of Sheridan Healthcare and future President of our combined Physician Services organization, and provide an overview of our future-state physician services offering.

When the merger is complete, our two physician services organizations — EmCare and Sheridan Healthcare — will join as one operating entity. The combination will create one of the largest, most diversified physician services enterprises in the industry, with expertise in more than two dozen specialties.

Such scope will enable us to provide more solutions across a broader continuum. As Bob conveys, as we grow together, we must also be equally committed to maintaining a culture that fosters innovation at the local level and creates strong alignment among our providers across all of our specialties.

Thank you for all you do,

If you have questions regarding the merger, please email EnvisionQuestions@evhc.net.

All merger updates can be found at www.evhc.net/merger.

[The following are transcripts of the videos sent with the above email to employees of AmSurg Corp. and Envision Healthcare Holdings, Inc. on October 27, 2016.]

Questions and Answers

With Bob Coward

[Bob]: I think this merger has amazing opportunities. What we have to do is remember one thing: keep the focus on the patient. Hi, I’m Bob Coward, and I’m the President of Sheridan Healthcare.

HOW DID YOU GET YOUR START IN HEALTHCARE?

[Bob]: You know, it was by accident. I had a brother and sister suffering from a chronic condition. And, I was offered an opportunity to work in a healthcare facility that treated those conditions, so I took that opportunity almost 26 years ago. And I’ve been in healthcare for over 26 years.

WHAT IS PHYSICIAN SERVICES?

[Bob]: That is a big question. I’ve always been a believer that if we truly want to see change in the healthcare delivery system, it starts with the physicians. And that’s what I want to do, is help our physicians and our other clinicians drive that change. Physician Services for me, and for what I think we have to do as a company, is really enabling physicians to do what they do best: treat patients, improve the care that we are giving our patients, and drive the change that we need in our healthcare delivery system.

WHAT IS THE MOST IMPORTANT THING TO YOU AS A PHYSICIAN SERVICES LEADER?

[Bob]: I think we have to focus on culture. Culture is the key to the success of any organization. And that starts with the first person that picks up a phone when a physician or clinician calls, all the way to the highest levels of our executive leadership. We have to understand and respect that our clinicians are the ones that are on the front lines and we need to be doing what we need to do to make their lives better every day.

HOW WILL OUR ORGANIZATION’S GROWTH IMPACT OUR TEAM MEMBERS AND PARTNERS?

[Bob]: That’s always the easiest question for me to answer. With growth comes opportunity, because with growth we need talent. And so the demands, the challenges, are always growing and expanding. And the best thing we have are the people that are already in the company, that are ready to grow and meet those challenges.

Physicians Services Overview

[Voiceover]: Upon completion of the AMSURG and Envision Healthcare merger, two outstanding physician services organizations will join forces.

[Voiceover]: EmCare, one of the nation’s leading providers of emergency medicine services and Sheridan Healthcare, a preeminent anesthesia provider.

[Voiceover]: The two organizations also have practices in many other in-hospital specialties and provide out-of hospital care too. No matter where the care is delivered — in-hospital, out-of-hospital or through

telemedicine — the physician services teams manage all operations of the practice, including staffing the clinicians, billing and all other administrative services.

[Voiceover]: Sheridan and EmCare both partner with hospitals and health systems to build solutions utilizing data-driven best practices, proprietary technology, process improvement and lean strategies so physicians and the clinicians who work alongside them can keep their focus on delivering the best possible patient care and quality outcomes.

[Voiceover]: Collectively, EmCare and Sheridan have more than 27,000 physicians, clinicians and support team members who registered more than 23 million patient encounters in 2015. That’s one patient every 81 seconds.

[Voiceover]: When merged, Sheridan and EmCare’s combined scale, scope and technology will further enable our top-notch clinicians and support teams to provide industry-leading solutions for our patients, partners and communities.

No Offer or Solicitation / Additional Information and Where to Find It

This communication is for informational purposes only and does not constitute an offer to sell or the solicitation of an offer to buy any securities or a solicitation of any vote or approval with respect to the proposed business combination between Envision Healthcare Holdings, Inc. (“Envision”) and AmSurg Corp. (“AMSURG”) or otherwise, nor shall there be any sale, issuance or transfer of securities in any jurisdiction in contravention of applicable law. No offer of securities shall be made except by means of a prospectus meeting the requirements of Section 10 of the Securities Act of 1933, as amended. The proposed business combination between Envision and AMSURG will be submitted to their respective shareholders on November 28, 2016 for consideration. On August 4, 2016, AMSURG caused its newly formed, wholly owned subsidiary, New Amethyst Corp. (“New Amethyst”), to file with the Securities and Exchange Commission (the “SEC”) a Registration Statement on Form S-4 (File No. 333-212885) that constitutes a prospectus of New Amethyst and a joint proxy statement of Envision and AMSURG. The SEC declared the Form S-4, as subsequently amended, effective on October 19, 2016.  Envision and AMSURG have caused the definitive joint proxy statement/final prospectus to be mailed to their respective shareholders, as required by applicable law. This communication is not a substitute for the definitive joint proxy statement/final prospectus, or any other document that may be filed with the SEC in connection with the proposed business combination. Investors and shareholders are urged to read carefully and in their entirety the definitive joint proxy statement/final prospectus delivered to shareholders, and any other relevant documents that are filed with the SEC when they become available, because they contain important information about the proposed business combination and related matters. Investors and shareholders may obtain free copies of the definitive joint proxy statement/final prospectus and other documents containing important information about Envision, AMSURG and New Amethyst, once such documents are filed with the SEC, through the website maintained by the SEC at www.sec.gov. Envision and AMSURG make available free of charge at www.evhc.net and www.amsurg.com, respectively (in the “Investors” section), copies of materials they file with, or furnish to, the SEC.

Participants in The Merger Solicitation

Envision, AMSURG and certain of their respective directors, executive officers and other members of management and employees may be deemed to be participants in the solicitation of proxies from the stockholders of Envision and shareholders of AMSURG in connection with the proposed business combination. Information about the directors and executive officers of Envision is set forth in its proxy statement for its 2016 annual meeting of stockholders filed with the SEC on March 23, 2016. Information about the directors and executive officers of AMSURG is set forth in its proxy statement for its 2016 annual meeting of shareholders filed with the SEC on April 22, 2016 and its Annual Report on Form 10-K for the year ended December 31, 2015 filed with the SEC on February 25, 2016. These documents can be obtained free of charge from the sources indicated above. Other information regarding those persons who are, under the rules of the SEC, participants in the proxy solicitation and a description of their direct and indirect interests, by security holdings or otherwise, are included in the definitive joint proxy statement/final prospectus.

Forward-Looking Statements

Certain statements and information in this communication may be deemed to be “forward-looking statements” within the meaning of the Federal Private Securities Litigation Reform Act of 1995. Forward-looking statements may include, but are not limited to, statements relating to Envision’s and AMSURG’s objectives, plans and strategies, and all statements (other than statements of historical facts) that address activities, events or developments that Envision and AMSURG intend, expect, project, believe or anticipate will or may occur in the future. These

statements are often characterized by terminology such as “believe,” “hope,” “may,” “anticipate,” “should,” “intend,” “plan,” “will,” “expect,” “estimate,” “project,” “positioned,” “strategy” and similar expressions, and are based on assumptions and assessments made by Envision’s and AMSURG’s management in light of their experience and their perception of historical trends, current conditions, expected future developments, and other factors they believe to be appropriate. Any forward-looking statements in this communication are made as of the date hereof, and Envision and AMSURG undertake no duty to update or revise any such statements, whether as a result of new information, future events or otherwise. Forward-looking statements are not guarantees of future performance. Whether actual results will conform to expectations and predictions is subject to known and unknown risks and uncertainties, including: (i) risks and uncertainties discussed in the reports that Envision and AMSURG have filed with the SEC; (ii) general economic, market, or business conditions; (iii) risks associated with the ability to consummate the business combination between Envision and AMSURG and the timing of the closing of the business combination; (iv) the ability to successfully integrate Envision’s and AMSURG’s operations and employees; (v) the ability to realize anticipated benefits and synergies of the business combination; (vi) the potential impact of announcement of the business combination or consummation of the transaction on relationships, including with employees, customers and competitors; and (vii) other circumstances beyond Envision’s and AMSURG’s control. Refer to the section entitled “Risk Factors” in Envision’s and AMSURG’s annual and quarterly reports for a discussion of important factors that could cause actual results, developments and business decisions to differ materially from forward-looking statements.