Theorem rexrab2 2846

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

Hypothesis

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

Assertion

Ref	Expression
rexrab2	|- ( E. x e. { y e. A | ph } ps <-> E. 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 rexrab2

Step	Hyp	Ref	Expression
1		df-rab 2423	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	rexeqi 2629	. 2 |- ( E. x e. { y e. A | ph } ps <-> E. x e. { y | ( y e. A /\ ph ) } ps )
3		ralab2.1	. . 3 |- ( x = y -> ( ps <-> ch ) )
4	3	rexab2 2845	. 2 |- ( E. x e. { y | ( y e. A /\ ph ) } ps <-> E. y ( ( y e. A /\ ph ) /\ ch ) )
5		anass 398	. . . 4 |- ( ( ( y e. A /\ ph ) /\ ch ) <-> ( y e. A /\ ( ph /\ ch ) ) )
6	5	exbii 1584	. . 3 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
7		df-rex 2420	. . 3 |- ( E. y e. A ( ph /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
8	6, 7	bitr4i 186	. 2 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y e. A ( ph /\ ch ) )
9	2, 4, 8	3bitri 205	1 |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1468   e. wcel 1480   {cab 2123   E.wrex 2415   {crab 2418

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-rab 2423

This theorem is referenced by: (None)