Theorem rspce 2872

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 2348	. . . 4 |- F/_ x A
2		nfv 1551	. . . . 5 |- F/ x A e. B
3		rspc.1	. . . . 5 |- F/ x ps
4	2, 3	nfan 1588	. . . 4 |- F/ x ( A e. B /\ ps )
5		eleq1 2268	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
6		rspc.2	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	anbi12d 473	. . . 4 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
8	1, 4, 7	spcegf 2856	. . 3 |- ( A e. B -> ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) ) )
9	8	anabsi5 579	. 2 |- ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) )
10		df-rex 2490	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
11	9, 10	sylibr 134	1 |- ( ( A e. B /\ ps ) -> E. x e. B ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1483   E.wex 1515   e. wcel 2176   E.wrex 2485

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774

This theorem is referenced by:  rspcev  2877  bezoutlemmain  12319