Whether or not the set that believes exists or not does not change the nature of the statement. It's still a conditional statement. In order for "whosoever" to obtain eternal life and avoid perishing, they must believe in The Son of God. This statement doesn't address who "whosoever" is, other than to make it open to anyone who believes. But the believing is the condition. One cannot avoid perishing and obtain Eternal life UNLESS one believes in the Son of God. That is an airtight condition.
Whether or not "whosoever"is all men or just a select set of men is not addressed, either positively or negatively in this statement. It neither establishes nor denies either Calvinist or Arminian doctrines with regard to Unconditional or Conditional Election, Limited or Unlimited Atonement. This passage, these two verses you quoted do not speak to those issues at all, and cannot be made to, either way.
These verses state a simple conditional statement. The question as to how belief arises or is applied is not addressed here. Belief is assumed, and made the condition which allows "whosoever" to obtain that which is promised.