Theorem rexeqf 2955

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 2574	. . 3 |- F/ x A = B
4		eleq2 2489	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 709	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	exbid 1941	. 2 |- ( A = B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
7		df-rex 2714	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2714	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6, 7, 8	3bitr4g 291	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   F/_wnfc 2550   E.wrex 2709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-ext 2402

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-cleq 2415  df-clel 2418  df-nfc 2552  df-rex 2714

This theorem is referenced by:  rexeq  2959  rexeqbid  2971  zfrep6  6712  iuneq12daf  28109  indexa  31961