Theorem rexab 2969

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 2517	. 2 |- ( E. x e. { y | ph } ch <-> E. x ( x e. { y | ph } /\ ch ) )
2		vex 2806	. . . . 5 |- x e. _V
3		ralab.1	. . . . 5 |- ( y = x -> ( ph <-> ps ) )
4	2, 3	elab 2951	. . . 4 |- ( x e. { y | ph } <-> ps )
5	4	anbi1i 458	. . 3 |- ( ( x e. { y | ph } /\ ch ) <-> ( ps /\ ch ) )
6	5	exbii 1654	. 2 |- ( E. x ( x e. { y | ph } /\ ch ) <-> E. x ( ps /\ ch ) )
7	1, 6	bitri 184	1 |- ( E. x e. { y | ph } ch <-> E. x ( ps /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1541   e. wcel 2202   {cab 2217   E.wrex 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805

This theorem is referenced by:  rexrnmpo  6147  4sqlem12  13038