Theorem reueq1f 2503

Description: Equality theorem for restricted uniqueness 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 2185	. . 3 |- F/ x A = B
4		eleq2 2101	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 438	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	eubid 1907	. 2 |- ( A = B -> ( E! x ( x e. A /\ ph ) <-> E! x ( x e. B /\ ph ) ) )
7		df-reu 2313	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
8		df-reu 2313	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
9	6, 7, 8	3bitr4g 212	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E!weu 1900   F/_wnfc 2165   E!wreu 2308

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-cleq 2033  df-clel 2036  df-nfc 2167  df-reu 2313

This theorem is referenced by:  reueq1  2507