Theorem raleqbidv 2562

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)

Hypotheses

Ref	Expression
raleqbidv.1	|- ( ph -> A = B )
raleqbidv.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
raleqbidv	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem raleqbidv

Step	Hyp	Ref	Expression
1		raleqbidv.1	. . 3 |- ( ph -> A = B )
2	1	raleqdv 2556	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )
3		raleqbidv.2	. . 3 |- ( ph -> ( ps <-> ch ) )
4	3	ralbidv 2369	. 2 |- ( ph -> ( A. x e. B ps <-> A. x e. B ch ) )
5	2, 4	bitrd 186	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   A.wral 2349

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354

This theorem is referenced by:  ofrfval  5745  fmpt2x  5851  tfrlemi1  5975  supeq123d  6453  cvg1nlemcau  9997  cvg1nlemres  9998  cau3lem  10127  sscoll2  10926