Theorem rexrab 2847

Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab.1	|- ( y = x -> ( ph <-> ps ) )

Assertion

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

Step	Hyp	Ref	Expression
1		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
2	1	elrab 2840	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	anbi1i 453	. . 3 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( ( x e. A /\ ps ) /\ ch ) )
4		anass 398	. . 3 |- ( ( ( x e. A /\ ps ) /\ ch ) <-> ( x e. A /\ ( ps /\ ch ) ) )
5	3, 4	bitri 183	. 2 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( x e. A /\ ( ps /\ ch ) ) )
6	5	rexbii2 2446	1 |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   E.wrex 2417   {crab 2420

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688

This theorem is referenced by: (None)