Theorem rspcegf 39182

Description: A version of rspcev 3309 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
rspcegf.1	|- F/ x ps
rspcegf.2	|- F/_ x A
rspcegf.3	|- F/_ x B
rspcegf.4	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem rspcegf

Step	Hyp	Ref	Expression
1		rspcegf.2	. . . 4 |- F/_ x A
2		rspcegf.3	. . . . . 6 |- F/_ x B
3	1, 2	nfel 2777	. . . . 5 |- F/ x A e. B
4		rspcegf.1	. . . . 5 |- F/ x ps
5	3, 4	nfan 1828	. . . 4 |- F/ x ( A e. B /\ ps )
6		eleq1 2689	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
7		rspcegf.4	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
8	6, 7	anbi12d 747	. . . 4 |- ( x = A -> ( ( x e. B /\ ph ) <-> ( A e. B /\ ps ) ) )
9	1, 5, 8	spcegf 3289	. . 3 |- ( A e. B -> ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) ) )
10	9	anabsi5 858	. 2 |- ( ( A e. B /\ ps ) -> E. x ( x e. B /\ ph ) )
11		df-rex 2918	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
12	10, 11	sylibr 224	1 |- ( ( A e. B /\ ps ) -> E. x e. B ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202

This theorem is referenced by:  rspcef  39241  stoweidlem46  40263