Theorem ralrab 2702

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 2698	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	imbi1i 227	. . 3 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( ( x e. A /\ ps ) -> ch ) )
4		impexp 250	. . 3 |- ( ( ( x e. A /\ ps ) -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
5	3, 4	bitri 173	. 2 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
6	5	ralbii2 2334	1 |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   A.wral 2306   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559

This theorem is referenced by: (None)