Theorem ralrab 2999

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 2995	. . . 4 |- (x e. {y e. A | ph} <-> (x e. A /\ ps))
3	2	imbi1i 315	. . 3 |- ((x e. {y e. A | ph} -> ch) <-> ((x e. A /\ ps) -> ch))
4		impexp 433	. . 3 |- (((x e. A /\ ps) -> ch) <-> (x e. A -> (ps -> ch)))
5	3, 4	bitri 240	. 2 |- ((x e. {y e. A | ph} -> ch) <-> (x e. A -> (ps -> ch)))
6	5	ralbii2 2643	1 |- (A.x e. {y e. A | ph}ch <-> A.x e. A (ps -> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   e. wcel 1710  A.wral 2615  {crab 2619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862

This theorem is referenced by: (None)