Theorem rexrab 3190

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 3184	. . . 4 |- ( x e. { y e. A | ph } <-> ( x e. A /\ ps ) )
3	2	anbi1i 709	. . 3 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( ( x e. A /\ ps ) /\ ch ) )
4		anass 661	. . 3 |- ( ( ( x e. A /\ ps ) /\ ch ) <-> ( x e. A /\ ( ps /\ ch ) ) )
5	3, 4	bitri 257	. 2 |- ( ( x e. { y e. A | ph } /\ ch ) <-> ( x e. A /\ ( ps /\ ch ) ) )
6	5	rexbii2 2879	1 |- ( E. x e. { y e. A | ph } ch <-> E. x e. A ( ps /\ ch ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   e. wcel 1904   E.wrex 2757   {crab 2760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033

This theorem is referenced by:  wereu2  4836  wdom2d  8113  enfin2i  8769  infm3  10590  pmtrfrn  17177  pgpssslw  17344  ellspd  19437  1stcfb  20537  xkobval  20678  xkococn  20752  imasdsf1olem  21466  nbgraf1olem1  25248  rusgranumwlks  25763  cvmliftlem15  30093  wsuclem  30579  poimirlem4  32008  poimirlem26  32030  poimirlem27  32031  rexrabdioph  35708  hbtlem6  36059