Theorem ralab 2920

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 2477	. 2 |- ( A. x e. { y | ph } ch <-> A. x ( x e. { y | ph } -> ch ) )
2		vex 2763	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2904	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	imbi1i 238	. . 3 |- ( ( x e. { y | ph } -> ch ) <-> ( ps -> ch ) )
6	5	albii 1481	. 2 |- ( A. x ( x e. { y | ph } -> ch ) <-> A. x ( ps -> ch ) )
7	1, 6	bitri 184	1 |- ( A. x e. { y | ph } ch <-> A. x ( ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   e. wcel 2164   {cab 2179   A.wral 2472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762

This theorem is referenced by:  funcnvuni  5323  ralrnmpo  6033  pitonn  7908