Theorem ralrab 2753

Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypothesis

Ref	Expression
ralab.1	|- ( y = x -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralrab	|- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )

Distinct variable groups:   x, y   y, A   ps, y

Allowed substitution hints:   ph( x, y)   ps( x)   ch( x, y)   A( x)

Proof of Theorem ralrab

Step	Hyp	Ref	Expression
1		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
2	1	elrab 2749	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	imbi1i 236	. . 3 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( ( x e. A /\ ps ) -> ch ) )
4		impexp 259	. . 3 |- ( ( ( x e. A /\ ps ) -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
5	3, 4	bitri 182	. 2 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
6	5	ralbii2 2376	1 |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   A.wral 2348   {crab 2352

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-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603

This theorem is referenced by: (None)