Theorem ralab 2773

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 2364	. 2 |- ( A. x e. { y | ph } ch <-> A. x ( x e. { y | ph } -> ch ) )
2		vex 2622	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2758	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	imbi1i 236	. . 3 |- ( ( x e. { y | ph } -> ch ) <-> ( ps -> ch ) )
6	5	albii 1404	. 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 1287   e. wcel 1438   {cab 2074   A.wral 2359

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621

This theorem is referenced by:  funcnvuni  5069  ralrnmpt2  5741  pitonn  7364