Theorem rspcimedv 2703

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
rspcimedv	|- ( ph -> ( ch -> E. x e. B ps ) )

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

Allowed substitution hint:   ps( x)

Proof of Theorem rspcimedv

Step	Hyp	Ref	Expression
1		rspcimdv.1	. . 3 |- ( ph -> A e. B )
2		simpr 108	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
3	2	eleq1d 2147	. . . . . 6 |- ( ( ph /\ x = A ) -> ( x e. B <-> A e. B ) )
4	3	biimprd 156	. . . . 5 |- ( ( ph /\ x = A ) -> ( A e. B -> x e. B ) )
5		rspcimedv.2	. . . . 5 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
6	4, 5	anim12d 328	. . . 4 |- ( ( ph /\ x = A ) -> ( ( A e. B /\ ch ) -> ( x e. B /\ ps ) ) )
7	1, 6	spcimedv 2684	. . 3 |- ( ph -> ( ( A e. B /\ ch ) -> E. x ( x e. B /\ ps ) ) )
8	1, 7	mpand 419	. 2 |- ( ph -> ( ch -> E. x ( x e. B /\ ps ) ) )
9		df-rex 2354	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
10	8, 9	syl6ibr 160	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603

This theorem is referenced by:  rspcedv  2705