Theorem rspcimedv 2870

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 110	. . . . . . 7 |- ( ( ph /\ x = A ) -> x = A )
3	2	eleq1d 2265	. . . . . 6 |- ( ( ph /\ x = A ) -> ( x e. B <-> A e. B ) )
4	3	biimprd 158	. . . . 5 |- ( ( ph /\ x = A ) -> ( A e. B -> x e. B ) )
5		rspcimedv.2	. . . . 5 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
6	4, 5	anim12d 335	. . . 4 |- ( ( ph /\ x = A ) -> ( ( A e. B /\ ch ) -> ( x e. B /\ ps ) ) )
7	1, 6	spcimedv 2850	. . 3 |- ( ph -> ( ( A e. B /\ ch ) -> E. x ( x e. B /\ ps ) ) )
8	1, 7	mpand 429	. 2 |- ( ph -> ( ch -> E. x ( x e. B /\ ps ) ) )
9		df-rex 2481	. 2 |- ( E. x e. B ps <-> E. x ( x e. B /\ ps ) )
10	8, 9	imbitrrdi 162	1 |- ( ph -> ( ch -> E. x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765

This theorem is referenced by:  rspcedv  2872