Theorem raleqf 2545

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 2226	. . 3 |- F/ x A = B
4		eleq2 2142	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
5	4	imbi1d 229	. . 3 |- ( A = B -> ( ( x e. A -> ph ) <-> ( x e. B -> ph ) ) )
6	3, 5	albid 1546	. 2 |- ( A = B -> ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ph ) ) )
7		df-ral 2353	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
8		df-ral 2353	. 2 |- ( A. x e. B ph <-> A. x ( x e. B -> ph ) )
9	6, 7, 8	3bitr4g 221	1 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   F/_wnfc 2206   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353

This theorem is referenced by:  raleq  2549  repizf2  3936