Theorem rexab 2775

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 2365	. 2 |- ( E. x e. { y | ph } ch <-> E. x ( x e. { y | ph } /\ ch ) )
2		vex 2622	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2758	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	anbi1i 446	. . 3 |- ( ( x e. { y | ph } /\ ch ) <-> ( ps /\ ch ) )
6	5	exbii 1541	. 2 |- ( E. x ( x e. { y | ph } /\ ch ) <-> E. x ( ps /\ ch ) )
7	1, 6	bitri 182	1 |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1426   e. wcel 1438   {cab 2074   E.wrex 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621

This theorem is referenced by:  rexrnmpt2  5742