Theorem rexrab2 2708

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 2315	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	rexeqi 2510	. 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 2707	. 2 |- ( E. x e. { y | ( y e. A /\ ph ) } ps <-> E. y ( ( y e. A /\ ph ) /\ ch ) )
5		anass 381	. . . 4 |- ( ( ( y e. A /\ ph ) /\ ch ) <-> ( y e. A /\ ( ph /\ ch ) ) )
6	5	exbii 1496	. . 3 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
7		df-rex 2312	. . 3 |- ( E. y e. A ( ph /\ ch ) <-> E. y ( y e. A /\ ( ph /\ ch ) ) )
8	6, 7	bitr4i 176	. 2 |- ( E. y ( ( y e. A /\ ph ) /\ ch ) <-> E. y e. A ( ph /\ ch ) )
9	2, 4, 8	3bitri 195	1 |- ( E. x e. { y e. A | ph } ps <-> E. y e. A ( ph /\ ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315

This theorem is referenced by: (None)