Theorem bj-spcimdv

Description: Remove from spcimdv dependency on ax-9 , ax-10 , ax-11 , ax-13 , ax-ext , df-cleq (and df-nfc , df-v , df-or , df-tru , df-nf ). For an even more economical version, see bj-spcimdvv . (Contributed by BJ, 30-Nov-2020) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-spcimdv.1	|- ( ph -> A e. B )
2		bj-spcimdv.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		bj-elisset	|- ( 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 ) )