Theorem ralab 2701

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 2311	. 2 |- ( A. x e. { y | ph } ch <-> A. x ( x e. { y | ph } -> ch ) )
2		vex 2560	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2687	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	imbi1i 227	. . 3 |- ( ( x e. { y | ph } -> ch ) <-> ( ps -> ch ) )
6	5	albii 1359	. 2 |- ( A. x ( x e. { y | ph } -> ch ) <-> A. x ( ps -> ch ) )
7	1, 6	bitri 173	1 |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   e. wcel 1393   {cab 2026   A.wral 2306

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-v 2559

This theorem is referenced by:  funcnvuni  4968  ralrnmpt2  5615  pitonn  6924