Theorem rmoeq1f 2651

Description: Equality theorem for restricted at-most-one quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypotheses

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

Assertion

Ref	Expression
rmoeq1f	|- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )

Proof of Theorem rmoeq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 |- F/_ x A
2		raleq1f.2	. . . 4 |- F/_ x B
3	1, 2	nfeq 2307	. . 3 |- F/ x A = B
4		eleq2 2221	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	anbi1d 461	. . 3 |- ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
6	3, 5	mobid 2041	. 2 |- ( A = B -> ( E* x ( x e. A /\ ph ) <-> E* x ( x e. B /\ ph ) ) )
7		df-rmo 2443	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
8		df-rmo 2443	. 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 1335   E*wmo 2007   e. wcel 2128   F/_wnfc 2286   E*wrmo 2438

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rmo 2443

This theorem is referenced by:  rmoeq1  2655