Theorem ralrab2 2873

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 2441	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	raleqi 2653	. 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 2872	. 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 1447	. . 3 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
7		df-ral 2437	. . 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 1330   e. wcel 2125   {cab 2140   A.wral 2432   {crab 2436

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rab 2441

This theorem is referenced by: (None)