Theorem rspcimedv 2658

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 103	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
3	2	eleq1d 2106	. . . . . 6 |- ( ( ph /\ x = A ) -> ( x e. B <-> A e. B ) )
4	3	biimprd 147	. . . . 5 |- ( ( ph /\ x = A ) -> ( A e. B -> x e. B ) )
5		rspcimedv.2	. . . . 5 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
6	4, 5	anim12d 318	. . . 4 |- ( ( ph /\ x = A ) -> ( ( A e. B /\ ch ) -> ( x e. B /\ ps ) ) )
7	1, 6	spcimedv 2639	. . 3 |- ( ph -> ( ( A e. B /\ ch ) -> E. x ( x e. B /\ ps ) ) )
8	1, 7	mpand 405	. 2 |- ( ph -> ( ch -> E. x ( x e. B /\ ps ) ) )
9		df-rex 2312	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
10	8, 9	syl6ibr 151	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559

This theorem is referenced by:  rspcedv  2660