Theorem rexrab2 2919

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- ( x = y -> ( ps <-> ch ) )

Assertion

Ref	Expression
rexrab2	|- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )

Distinct variable groups:   x, y   x, A   ch, x   ph, x   ps, y

Allowed substitution hints:   ph( y)   ps( x)   ch( y)   A( y)

Proof of Theorem rexrab2

Step	Hyp	Ref	Expression
1		df-rab 2477	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	rexeqi 2691	. 2 |- ( E. x e. { y e. A | ph } ps <-> E. x e. { y | ( y e. A /\ ph ) } ps )
3		ralab2.1	. . 3 |- ( x = y -> ( ps <-> ch ) )
4	3	rexab2 2918	. 2 |- ( E. x e. { y | ( y e. A /\ ph ) } ps <-> E. y ( ( y e. A /\ ph ) /\ ch ) )
5		anass 401	. . . 4 |- ( ( ( y e. A /\ ph ) /\ ch ) <-> ( y e. A /\ ( ph /\ ch ) ) )
6	5	exbii 1616	. . 3 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
7		df-rex 2474	. . 3 |- ( E. y e. A ( ph /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
8	6, 7	bitr4i 187	. 2 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y e. A ( ph /\ ch ) )
9	2, 4, 8	3bitri 206	1 |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1503   e. wcel 2160   {cab 2175   E.wrex 2469   {crab 2472

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-rab 2477

This theorem is referenced by: (None)