Theorem bj-spcimdvv 32885

Description: Remove from spcimdv 3290 dependency on ax-7 1935, ax-8 1992, ax-10 2019, ax-11 2034, ax-12 2047 ax-13 2246, ax-ext 2602, df-cleq 2615, df-clab 2609 (and df-nfc 2753, df-v 3202, df-or 385, df-tru 1486, df-nf 1710) at the price of adding a DV condition on x , B (but in usages, x is typically a dummy, hence fresh, variable). For the version without this DV condition, see bj-spcimdv 32884. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-spcimdvv	|- ( ph -> ( A. x ps -> ch ) )

Distinct variable groups:   x, A   x, B   ph, x   ch, x

Allowed substitution hint:   ps( x)

Proof of Theorem bj-spcimdvv

Step	Hyp	Ref	Expression
1		bj-spcimdvv.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
2	1	ex 450	. . 3 |- ( ph -> ( x = A -> ( ps -> ch ) ) )
3	2	alrimiv 1855	. 2 |- ( ph -> A. x ( x = A -> ( ps -> ch ) ) )
4		bj-spcimdvv.1	. 2 |- ( ph -> A e. B )
5		bj-elissetv 32861	. . . 4 |- ( A e. B -> E. x x = A )
6		exim 1761	. . . 4 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( E. x x = A -> E. x ( ps -> ch ) ) )
7	5, 6	syl5 34	. . 3 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> E. x ( ps -> ch ) ) )
8		19.36v 1904	. . 3 |- ( E. x ( ps -> ch ) <-> ( A. x ps -> ch ) )
9	7, 8	syl6ib 241	. 2 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> ( A. x ps -> ch ) ) )
10	3, 4, 9	sylc 65	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-clel 2618

This theorem is referenced by: (None)