Theorem rexab 2841

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

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

Assertion

Ref	Expression
rexab	|- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

Distinct variable groups:   x, y   ps, y

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

Proof of Theorem rexab

Step	Hyp	Ref	Expression
1		df-rex 2420	. 2 |- ( E. x e. { y | ph } ch <-> E. x ( x e. { y | ph } /\ ch ) )
2		vex 2684	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2823	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	anbi1i 453	. . 3 |- ( ( x e. { y | ph } /\ ch ) <-> ( ps /\ ch ) )
6	5	exbii 1584	. 2 |- ( E. x ( x e. { y | ph } /\ ch ) <-> E. x ( ps /\ ch ) )
7	1, 6	bitri 183	1 |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

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

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-v 2683

This theorem is referenced by:  rexrnmpo  5879