Theorem rspce 2718

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 2229	. . . 4 |- F/_ x A
2		nfv 1467	. . . . 5 |- F/ x A e. B
3		rspc.1	. . . . 5 |- F/ x ps
4	2, 3	nfan 1503	. . . 4 |- F/ x ( A e. B /\ ps )
5		eleq1 2151	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
6		rspc.2	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	anbi12d 458	. . . 4 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
8	1, 4, 7	spcegf 2703	. . 3 |- ( A e. B -> ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) ) )
9	8	anabsi5 547	. 2 |- ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) )
10		df-rex 2366	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
11	9, 10	sylibr 133	1 |- ( ( A e. B /\ ps ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   F/wnf 1395   E.wex 1427   e. wcel 1439   E.wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622

This theorem is referenced by:  rspcev  2723  bezoutlemmain  11326