Theorem bj-ax12ig

Description: A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i . (Contributed by BJ, 19-Dec-2020)

	Ref	Expression
Hypotheses	bj-ax12ig.1	|- ( ph -> ( ps <-> ch ) )
	bj-ax12ig.2	|- ( ph -> ( ch -> A. x ch ) )
Assertion	bj-ax12ig	|- ( ph -> ( ps -> A. x ( ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ax12ig.1	|- ( ph -> ( ps <-> ch ) )
2		bj-ax12ig.2	|- ( ph -> ( ch -> A. x ch ) )
3	1	pm5.32i	|- ( ( ph /\ ps ) <-> ( ph /\ ch ) )
4	2	imp	|- ( ( ph /\ ch ) -> A. x ch )
5	1	biimprcd	|- ( ch -> ( ph -> ps ) )
6	4 5	sylg	|- ( ( ph /\ ch ) -> A. x ( ph -> ps ) )
7	3 6	sylbi	|- ( ( ph /\ ps ) -> A. x ( ph -> ps ) )
8	7	ex	|- ( ph -> ( ps -> A. x ( ph -> ps ) ) )