Theorem ceqsrexbv 2870

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)

Hypothesis

Ref	Expression
ceqsrexv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem ceqsrexbv

Step	Hyp	Ref	Expression
1		r19.42v 2634	. 2 |- ( E. x e. B ( A e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ E. x e. B ( x = A /\ ph ) ) )
2		eleq1 2240	. . . . . . 7 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 276	. . . . . 6 |- ( ( x = A /\ ph ) -> ( x e. B <-> A e. B ) )
4	3	pm5.32ri 455	. . . . 5 |- ( ( x e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ ( x = A /\ ph ) ) )
5	4	bicomi 132	. . . 4 |- ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x e. B /\ ( x = A /\ ph ) ) )
6	5	baib 919	. . 3 |- ( x e. B -> ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x = A /\ ph ) ) )
7	6	rexbiia 2492	. 2 |- ( E. x e. B ( A e. B /\ ( x = A /\ ph ) ) <-> E. x e. B ( x = A /\ ph ) )
8		ceqsrexv.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
9	8	ceqsrexv 2869	. . 3 |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )
10	9	pm5.32i 454	. 2 |- ( ( A e. B /\ E. x e. B ( x = A /\ ph ) ) <-> ( A e. B /\ ps ) )
11	1, 7, 10	3bitr3i 210	1 |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741

This theorem is referenced by:  frecsuclem  6409