Theorem raleqf 2661

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 2320	. . 3 |- F/ x A = B
4		eleq2 2234	. . . 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 1608	. 2 |- ( A = B -> ( A. x ( x e. A -> ph ) <-> A. x ( x e. B -> ph ) ) )
7		df-ral 2453	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
8		df-ral 2453	. 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 1346   = wceq 1348   e. wcel 2141   F/_wnfc 2299   A.wral 2448

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453

This theorem is referenced by:  raleq  2665  repizf2  4148  ellimc3apf  13423