Metamath Proof Explorer


Theorem bj-nnfim2

Description: A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023)

Ref Expression
Assertion bj-nnfim2
|- ( ( F// x ph /\ F// x ps ) -> ( ( A. x ph -> E. x ps ) -> ( ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 bj-nnfa
 |-  ( F// x ph -> ( ph -> A. x ph ) )
2 bj-nnfe
 |-  ( F// x ps -> ( E. x ps -> ps ) )
3 imim12
 |-  ( ( ph -> A. x ph ) -> ( ( E. x ps -> ps ) -> ( ( A. x ph -> E. x ps ) -> ( ph -> ps ) ) ) )
4 3 imp
 |-  ( ( ( ph -> A. x ph ) /\ ( E. x ps -> ps ) ) -> ( ( A. x ph -> E. x ps ) -> ( ph -> ps ) ) )
5 1 2 4 syl2an
 |-  ( ( F// x ph /\ F// x ps ) -> ( ( A. x ph -> E. x ps ) -> ( ph -> ps ) ) )