Theorem ralrab 2898

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 2893	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	imbi1i 238	. . 3 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( ( x e. A /\ ps ) -> ch ) )
4		impexp 263	. . 3 |- ( ( ( x e. A /\ ps ) -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
5	3, 4	bitri 184	. 2 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
6	5	ralbii2 2487	1 |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   A.wral 2455   {crab 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739

This theorem is referenced by:  mhmeql  12830  limcdifap  14024