Theorem abexssex 6104

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 2454	. . . 4 |- ( E. x e. ~P A ph <-> E. x ( x e. ~P A /\ ph ) )
2		velpw 3573	. . . . . 6 |- ( x e. ~P A <-> x C_ A )
3	2	anbi1i 455	. . . . 5 |- ( ( x e. ~P A /\ ph ) <-> ( x C_ A /\ ph ) )
4	3	exbii 1598	. . . 4 |- ( E. x ( x e. ~P A /\ ph ) <-> E. x ( x C_ A /\ ph ) )
5	1, 4	bitri 183	. . 3 |- ( E. x e. ~P A ph <-> E. x ( x C_ A /\ ph ) )
6	5	abbii 2286	. 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 4169	. . 3 |- ~P A e. _V
9		abrexex2.2	. . 3 |- { y | ph } e. _V
10	8, 9	abrexex2 6103	. 2 |- { y | E. x e. ~P A ph } e. _V
11	6, 10	eqeltrri 2244	1 |- { y | E. x ( x C_ A /\ ph ) } e. _V

Syntax hints:   /\ wa 103   E.wex 1485   e. wcel 2141   {cab 2156   E.wrex 2449   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by: (None)