Theorem bj-nnfim1

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

	Ref	Expression
Assertion	bj-nnfim1	|- ( ( F// x ph /\ F// x ps ) -> ( ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )

Proof

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