Theorem raleqf 2657

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

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

Assertion

Ref	Expression
raleqf	|- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )

Proof of Theorem raleqf

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 |- F/_ x A
2		raleq1f.2	. . . 4 |- F/_ x B
3	1, 2	nfeq 2316	. . 3 |- F/ x A = B
4		eleq2 2230	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	imbi1d 230	. . 3 |- ( A = B -> ( ( x e. A -> ph ) <-> ( x e. B -> ph ) ) )
6	3, 5	albid 1603	. 2 |- ( A = B -> ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ph ) ) )
7		df-ral 2449	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
8		df-ral 2449	. 2 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
9	6, 7, 8	3bitr4g 222	1 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   F/_wnfc 2295   A.wral 2444

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by:  raleq  2661  repizf2  4141  ellimc3apf  13269