Theorem rexeqf 3055

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
raleq1f.1	|- F/_ x A
raleq1f.2	|- F/_ x B

Assertion

Ref	Expression
rexeqf	|- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Proof of Theorem rexeqf

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 |- F/_ x A
2		raleq1f.2	. . . 4 |- F/_ x B
3	1, 2	nfeq 2640	. . 3 |- F/ x A = B
4		eleq2 2540	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 704	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	exbid 1834	. 2 |- ( A = B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
7		df-rex 2820	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2820	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6, 7, 8	3bitr4g 288	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   F/_wnfc 2615   E.wrex 2815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820

This theorem is referenced by:  rexeq  3059  rexeqbid  3071  zfrep6  6752  iuneq12daf  27114  indexa  29843