Theorem rexeqf 2805

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/_xA
raleq1f.2	|- F/_xB

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/_xA
2		raleq1f.2	. . . 4 |- F/_xB
3	1, 2	nfeq 2497	. . 3 |- F/x A = B
4		eleq2 2414	. . . 4 |- (A = B -> (x e. A <-> x e. B))
5	4	anbi1d 685	. . 3 |- (A = B -> ((x e. A /\ ph) <-> (x e. B /\ ph)))
6	3, 5	exbid 1773	. 2 |- (A = B -> (E.x(x e. A /\ ph) <-> E.x(x e. B /\ ph)))
7		df-rex 2621	. 2 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
8		df-rex 2621	. 2 |- (E.x e. B ph <-> E.x(x e. B /\ ph))
9	6, 7, 8	3bitr4g 279	1 |- (A = B -> (E.x e. A ph <-> E.x e. B ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   F/_wnfc 2477  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by:  rexeq  2809