Theorem ralab 2752

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

Hypothesis

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

Assertion

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

Distinct variable groups:   x, y   ps, y

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

Proof of Theorem ralab

Step	Hyp	Ref	Expression
1		df-ral 2353	. 2 |- ( A. x e. { y | ph } ch <-> A. x ( x e. { y | ph } -> ch ) )
2		vex 2604	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2738	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	imbi1i 236	. . 3 |- ( ( x e. { y | ph } -> ch ) <-> ( ps -> ch ) )
6	5	albii 1399	. 2 |- ( A. x ( x e. { y | ph } -> ch ) <-> A. x ( ps -> ch ) )
7	1, 6	bitri 182	1 |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   A.wral 2348

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-v 2603

This theorem is referenced by:  funcnvuni  4988  ralrnmpt2  5635  pitonn  7016