Theorem ralrab 2887

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 2882	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	imbi1i 237	. . 3 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( ( x e. A /\ ps ) -> ch ) )
4		impexp 261	. . 3 |- ( ( ( x e. A /\ ps ) -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
5	3, 4	bitri 183	. 2 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
6	5	ralbii2 2476	1 |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   A.wral 2444   {crab 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 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-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728

This theorem is referenced by:  limcdifap  13271