Theorem ceqsrexbv 2675

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 2467	. 2 |- ( E. x e. B ( A e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ E. x e. B ( x = A /\ ph ) ) )
2		eleq1 2100	. . . . . . 7 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 261	. . . . . 6 |- ( ( x = A /\ ph ) -> ( x e. B <-> A e. B ) )
4	3	pm5.32ri 428	. . . . 5 |- ( ( x e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ ( x = A /\ ph ) ) )
5	4	bicomi 123	. . . 4 |- ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x e. B /\ ( x = A /\ ph ) ) )
6	5	baib 828	. . 3 |- ( x e. B -> ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x = A /\ ph ) ) )
7	6	rexbiia 2339	. 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 2674	. . 3 |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )
10	9	pm5.32i 427	. 2 |- ( ( A e. B /\ E. x e. B ( x = A /\ ph ) ) <-> ( A e. B /\ ps ) )
11	1, 7, 10	3bitr3i 199	1 |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559

This theorem is referenced by:  frecsuclem3  5990