Theorem bj-spcimdvv

Description: Remove from spcimdv dependency on ax-7 , ax-8 , ax-10 , ax-11 , ax-12 ax-13 , ax-ext , df-cleq , df-clab (and df-nfc , df-v , df-or , df-tru , df-nf ) at the price of adding a disjoint variable condition on x , B (but in usages, x is typically a dummy, hence fresh, variable). For the version without this disjoint variable condition, see bj-spcimdv . (Contributed by BJ, 3-Nov-2021) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-spcimdvv.1	|- ( ph -> A e. B )
2		bj-spcimdvv.2	|- ( ( ph /\ x = A ) -> ( ps -> ch ) )
3	2	ex	|- ( ph -> ( x = A -> ( ps -> ch ) ) )
4	3	alrimiv	|- ( ph -> A. x ( x = A -> ( ps -> ch ) ) )
5		elissetv	|- ( A e. B -> E. x x = A )
6		exim	|- ( A. x ( x = A -> ( ps -> ch ) ) -> ( E. x x = A -> E. x ( ps -> ch ) ) )
7	5 6	syl5	|- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> E. x ( ps -> ch ) ) )
8		19.36v	|- ( E. x ( ps -> ch ) <-> ( A. x ps -> ch ) )
9	7 8	syl6ib	|- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> ( A. x ps -> ch ) ) )
10	4 1 9	sylc	|- ( ph -> ( A. x ps -> ch ) )