"经检验证明无病的" can be translated into English as "proven disease-free through examination" or "verified to be free of illness after inspection".
Both expressions convey the meaning that someone has undergone an examination or inspection and has been found not to have any disease or illness. The choice between them may depend on the specific context or the level of formality required.
"Proven disease-free through examination" emphasizes the process of examination and the resulting proof of being disease-free.
"Verified to be free of illness after inspection" highlights the verification process and the subsequent confirmation of being illness-free.
In most cases, either expression would be appropriate and understandable.