Theorem rspce 2951

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/xps
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 2490	. . . 4 |- F/_xA
2		nfv 1619	. . . . 5 |- F/x A e. B
3		rspc.1	. . . . 5 |- F/xps
4	2, 3	nfan 1824	. . . 4 |- F/x(A e. B /\ ps)
5		eleq1 2413	. . . . 5 |- (x = A -> (x e. B <-> A e. B))
6		rspc.2	. . . . 5 |- (x = A -> (ph <-> ps))
7	5, 6	anbi12d 691	. . . 4 |- (x = A -> ((x e. B /\ ph) <-> (A e. B /\ ps)))
8	1, 4, 7	spcegf 2936	. . 3 |- (A e. B -> ((A e. B /\ ps) -> E.x(x e. B /\ ph)))
9	8	anabsi5 790	. 2 |- ((A e. B /\ ps) -> E.x(x e. B /\ ph))
10		df-rex 2621	. 2 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
11	9, 10	sylibr 203	1 |- ((A e. B /\ ps) -> E.x e. B ph)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862

This theorem is referenced by:  rspcev  2956