Theorem rexrab2 2780

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 2368	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	rexeqi 2567	. 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 2779	. 2 |- ( E. x e. { y | ( y e. A /\ ph ) } ps <-> E. y ( ( y e. A /\ ph ) /\ ch ) )
5		anass 393	. . . 4 |- ( ( ( y e. A /\ ph ) /\ ch ) <-> ( y e. A /\ ( ph /\ ch ) ) )
6	5	exbii 1541	. . 3 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
7		df-rex 2365	. . 3 |- ( E. y e. A ( ph /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
8	6, 7	bitr4i 185	. 2 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y e. A ( ph /\ ch ) )
9	2, 4, 8	3bitri 204	1 |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1426   e. wcel 1438   {cab 2074   E.wrex 2360   {crab 2363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-rab 2368

This theorem is referenced by: (None)