Metamath Proof Explorer


Theorem bj-exlimg

Description: The general form of the *exlim* family of theorems: if ph is substituted for ps , then the antecedent expresses a form of nonfreeness of x in ph , so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg . (Contributed by BJ, 9-Dec-2023)

Ref Expression
Assertion bj-exlimg
|- ( ( E. x ph -> ps ) -> ( A. x ( ch -> ph ) -> ( E. x ch -> ps ) ) )

Proof

Step Hyp Ref Expression
1 bj-sylget
 |-  ( A. x ( ch -> ph ) -> ( ( E. x ph -> ps ) -> ( E. x ch -> ps ) ) )
2 1 com12
 |-  ( ( E. x ph -> ps ) -> ( A. x ( ch -> ph ) -> ( E. x ch -> ps ) ) )