Theorem ralrab2 2877

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- ( x = y -> ( ps <-> ch ) )

Assertion

Ref	Expression
ralrab2	|- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )

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

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

Proof of Theorem ralrab2

Step	Hyp	Ref	Expression
1		df-rab 2444	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	raleqi 2656	. 2 |- ( A. x e. { y e. A | ph } ps <-> A. x e. { y | ( y e. A /\ ph ) } ps )
3		ralab2.1	. . 3 |- ( x = y -> ( ps <-> ch ) )
4	3	ralab2 2876	. 2 |- ( A. x e. { y | ( y e. A /\ ph ) } ps <-> A. y ( ( y e. A /\ ph ) -> ch ) )
5		impexp 261	. . . 4 |- ( ( ( y e. A /\ ph ) -> ch ) <-> ( y e. A -> ( ph -> ch ) ) )
6	5	albii 1450	. . 3 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
7		df-ral 2440	. . 3 |- ( A. y e. A ( ph -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
8	6, 7	bitr4i 186	. 2 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y e. A ( ph -> ch ) )
9	2, 4, 8	3bitri 205	1 |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   e. wcel 2128   {cab 2143   A.wral 2435   {crab 2439

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444

This theorem is referenced by: (None)