Theorem spcimdv

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023)

	Ref	Expression
Hypotheses	spcimdv.1	|- ( ph -> A e. B )
	spcimdv.2	|- ( ( ph /\ x = A ) -> ( ps -> ch ) )
Assertion	spcimdv	|- ( ph -> ( A. x ps -> ch ) )

Proof

Step	Hyp	Ref	Expression
1		spcimdv.1	|- ( ph -> A e. B )
2		spcimdv.2	|- ( ( ph /\ x = A ) -> ( ps -> ch ) )
3		elisset	|- ( A e. B -> E. x x = A )
4	1 3	syl	|- ( ph -> E. x x = A )
5	2	ex	|- ( ph -> ( x = A -> ( ps -> ch ) ) )
6	5	eximdv	|- ( ph -> ( E. x x = A -> E. x ( ps -> ch ) ) )
7	4 6	mpd	|- ( ph -> E. x ( ps -> ch ) )
8		19.36v	|- ( E. x ( ps -> ch ) <-> ( A. x ps -> ch ) )
9	7 8	sylib	|- ( ph -> ( A. x ps -> ch ) )