Theorem rspce 2696

Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
rspc.1	|- F/ x ps
rspc.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rspce	|- ( ( A e. B /\ ps ) -> E. x e. B ph )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rspce

Step	Hyp	Ref	Expression
1		nfcv 2219	. . . 4 |- F/_ x A
2		nfv 1461	. . . . 5 |- F/ x A e. B
3		rspc.1	. . . . 5 |- F/ x ps
4	2, 3	nfan 1497	. . . 4 |- F/ x ( A e. B /\ ps )
5		eleq1 2141	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
6		rspc.2	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	anbi12d 456	. . . 4 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
8	1, 4, 7	spcegf 2681	. . 3 |- ( A e. B -> ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) ) )
9	8	anabsi5 543	. 2 |- ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) )
10		df-rex 2354	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
11	9, 10	sylibr 132	1 |- ( ( A e. B /\ ps ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   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:  rspcev  2701  bezoutlemmain  10387