Theorem rexab 2850

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 2423	. 2 |- ( E. x e. { y | ph } ch <-> E. x ( x e. { y | ph } /\ ch ) )
2		vex 2692	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2832	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	anbi1i 454	. . 3 |- ( ( x e. { y | ph } /\ ch ) <-> ( ps /\ ch ) )
6	5	exbii 1585	. 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 1469   e. wcel 1481   {cab 2126   E.wrex 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691

This theorem is referenced by:  rexrnmpo  5894