Theorem ralab 2886

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 2449	. 2 |- ( A. x e. { y | ph } ch <-> A. x ( x e. { y | ph } -> ch ) )
2		vex 2729	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2870	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	imbi1i 237	. . 3 |- ( ( x e. { y | ph } -> ch ) <-> ( ps -> ch ) )
6	5	albii 1458	. 2 |- ( A. x ( x e. { y | ph } -> ch ) <-> A. x ( ps -> ch ) )
7	1, 6	bitri 183	1 |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   e. wcel 2136   {cab 2151   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728

This theorem is referenced by:  funcnvuni  5257  ralrnmpo  5956  pitonn  7789