Theorem rexrab 3267

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 3261	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	anbi1i 695	. . 3 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( ( x e. A /\ ps ) /\ ch ) )
4		anass 649	. . 3 |- ( ( ( x e. A /\ ps ) /\ ch ) <-> ( x e. A /\ ( ps /\ ch ) ) )
5	3, 4	bitri 249	. 2 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( x e. A /\ ( ps /\ ch ) ) )
6	5	rexbii2 2963	1 |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   E.wrex 2815   {crab 2818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115

This theorem is referenced by:  wereu2  4876  wdom2d  8002  enfin2i  8697  infm3  10498  pmtrfrn  16276  pgpssslw  16427  ellspd  18600  ellspdOLD  18601  1stcfb  19709  xkobval  19819  xkococn  19893  imasdsf1olem  20608  nbgraf1olem1  24114  rusgranumwlks  24629  cvmliftlem15  28380  wsuclem  28955  rexrabdioph  30329  hbtlem6  30682