Theorem ceqsrexbv 2861

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 2627	. 2 |- ( E. x e. B ( A e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ E. x e. B ( x = A /\ ph ) ) )
2		eleq1 2233	. . . . . . 7 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 274	. . . . . 6 |- ( ( x = A /\ ph ) -> ( x e. B <-> A e. B ) )
4	3	pm5.32ri 452	. . . . 5 |- ( ( x e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ ( x = A /\ ph ) ) )
5	4	bicomi 131	. . . 4 |- ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x e. B /\ ( x = A /\ ph ) ) )
6	5	baib 914	. . 3 |- ( x e. B -> ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x = A /\ ph ) ) )
7	6	rexbiia 2485	. 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 2860	. . 3 |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )
10	9	pm5.32i 451	. 2 |- ( ( A e. B /\ E. x e. B ( x = A /\ ph ) ) <-> ( A e. B /\ ps ) )
11	1, 7, 10	3bitr3i 209	1 |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732

This theorem is referenced by:  frecsuclem  6385