Theorem reueq1f 2622

Description: Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

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

Assertion

Ref	Expression
reueq1f	|- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Proof of Theorem reueq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 |- F/_ x A
2		raleq1f.2	. . . 4 |- F/_ x B
3	1, 2	nfeq 2287	. . 3 |- F/ x A = B
4		eleq2 2201	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 460	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	eubid 2004	. 2 |- ( A = B -> ( E! x ( x e. A /\ ph ) <-> E! x ( x e. B /\ ph ) ) )
7		df-reu 2421	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
8		df-reu 2421	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
9	6, 7, 8	3bitr4g 222	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E!weu 1997   F/_wnfc 2266   E!wreu 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-cleq 2130  df-clel 2133  df-nfc 2268  df-reu 2421

This theorem is referenced by:  reueq1  2626