Theorem rexrab 3212

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 3206	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	anbi1i 693	. . 3 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( ( x e. A /\ ps ) /\ ch ) )
4		anass 647	. . 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 2903	1 |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   e. wcel 1842   E.wrex 2754   {crab 2757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060

This theorem is referenced by:  wereu2  4819  wdom2d  7960  enfin2i  8653  infm3  10462  pmtrfrn  16699  pgpssslw  16850  ellspd  19021  1stcfb  20130  xkobval  20271  xkococn  20345  imasdsf1olem  21060  nbgraf1olem1  24739  rusgranumwlks  25254  cvmliftlem15  29476  wsuclem  30054  rexrabdioph  35070  hbtlem6  35423