Theorem rexeqf 3135

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 2776	. . 3 |- F/ x A = B
4		eleq2 2690	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 741	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	exbid 2091	. 2 |- ( A = B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
7		df-rex 2918	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2918	. 2 |- ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
9	6, 7, 8	3bitr4g 303	1 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   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-11 2034  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-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918

This theorem is referenced by:  rexeq  3139  rexeqbid  3151  zfrep6  7134  iuneq12daf  29373  indexa  33528