Theorem ralrab 2925

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

Hypothesis

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

Assertion

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

Distinct variable groups:   x, y   y, A   ps, y

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

Proof of Theorem ralrab

Step	Hyp	Ref	Expression
1		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
2	1	elrab 2920	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	imbi1i 238	. . 3 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( ( x e. A /\ ps ) -> ch ) )
4		impexp 263	. . 3 |- ( ( ( x e. A /\ ps ) -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
5	3, 4	bitri 184	. 2 |- ( ( x e. { y e. A | ph } -> ch ) <-> ( x e. A -> ( ps -> ch ) ) )
6	5	ralbii2 2507	1 |- ( A. x e. { y e. A | ph } ch <-> A. x e. A ( ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2167   A.wral 2475   {crab 2479

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765

This theorem is referenced by:  mhmeql  13124  ghmeql  13397  limcdifap  14898