Theorem reueq1f 2747

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 2400	. . 3 |- F/ x A = B
4		eleq2 2302	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 469	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	eubid 2093	. 2 |- ( A = B -> ( E! x ( x e. A /\ ph ) <-> E! x ( x e. B /\ ph ) ) )
7		df-reu 2535	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
8		df-reu 2535	. 2 |- ( E! x e. B ph <-> E! x ( x e. B /\ ph ) )
9	6, 7, 8	3bitr4g 223	1 |- ( A = B -> ( E! x e. A ph <-> E! x e. B ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E!weu 2086   e. wcel 2209   F/_wnfc 2379   E!wreu 2530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-cleq 2231  df-clel 2234  df-nfc 2381  df-reu 2535

This theorem is referenced by:  reueq1  2751