Theorem ralrab2 2895

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 2457	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	raleqi 2669	. 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 2894	. 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 1463	. . 3 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
7		df-ral 2453	. . 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 1346   e. wcel 2141   {cab 2156   A.wral 2448   {crab 2452

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457

This theorem is referenced by: (None)