Theorem ceqsrexbv 3337

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 3092	. 2 |- ( E. x e. B ( A e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ E. x e. B ( x = A /\ ph ) ) )
2		eleq1 2689	. . . . . . 7 |- ( x = A -> ( x e. B <-> A e. B ) )
3	2	adantr 481	. . . . . 6 |- ( ( x = A /\ ph ) -> ( x e. B <-> A e. B ) )
4	3	pm5.32ri 670	. . . . 5 |- ( ( x e. B /\ ( x = A /\ ph ) ) <-> ( A e. B /\ ( x = A /\ ph ) ) )
5	4	bicomi 214	. . . 4 |- ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x e. B /\ ( x = A /\ ph ) ) )
6	5	baib 944	. . 3 |- ( x e. B -> ( ( A e. B /\ ( x = A /\ ph ) ) <-> ( x = A /\ ph ) ) )
7	6	rexbiia 3040	. 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 3336	. . 3 |- ( A e. B -> ( E. x e. B ( x = A /\ ph ) <-> ps ) )
10	9	pm5.32i 669	. 2 |- ( ( A e. B /\ E. x e. B ( x = A /\ ph ) ) <-> ( A e. B /\ ps ) )
11	1, 7, 10	3bitr3i 290	1 |- ( E. x e. B ( x = A /\ ph ) <-> ( A e. B /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   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-12 2047  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-rex 2918  df-v 3202

This theorem is referenced by:  marypha2lem2  8342  txkgen  21455  ceqsrexv2  31605  eq0rabdioph  37340