Theorem abexssex 6296

Description: Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in ph. (Contributed by NM, 29-Jul-2006.)

Hypotheses

Ref	Expression
abrexex2.1	|- A e. _V
abrexex2.2	|- { y | ph } e. _V

Assertion

Ref	Expression
abexssex	|- { y | E. x ( x C_ A /\ ph ) } e. _V

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem abexssex

Step	Hyp	Ref	Expression
1		df-rex 2517	. . . 4 |- ( E. x e. ~P A ph <-> E. x ( x e. ~P A /\ ph ) )
2		velpw 3663	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
3	2	anbi1i 458	. . . . 5 |- ( ( x e. ~P A /\ ph ) <-> ( x C_ A /\ ph ) )
4	3	exbii 1654	. . . 4 |- ( E. x ( x e. ~P A /\ ph ) <-> E. x ( x C_ A /\ ph ) )
5	1, 4	bitri 184	. . 3 |- ( E. x e. ~P A ph <-> E. x ( x C_ A /\ ph ) )
6	5	abbii 2347	. 2 |- { y | E. x e. ~P A ph } = { y | E. x ( x C_ A /\ ph ) }
7		abrexex2.1	. . . 4 |- A e. _V
8	7	pwex 4279	. . 3 |- ~P A e. _V
9		abrexex2.2	. . 3 |- { y | ph } e. _V
10	8, 9	abrexex2 6295	. 2 |- { y | E. x e. ~P A ph } e. _V
11	6, 10	eqeltrri 2305	1 |- { y | E. x ( x C_ A /\ ph ) } e. _V

Syntax hints:   /\ wa 104   E.wex 1541   e. wcel 2202   {cab 2217   E.wrex 2512   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341

This theorem is referenced by: (None)