Theorem abexex 6319

Description: A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Hypotheses

Ref	Expression
abexex.1	|- A e. _V
abexex.2	|- ( ph -> x e. A )
abexex.3	|- { y | ph } e. _V

Assertion

Ref	Expression
abexex	|- { y | E. x ph } e. _V

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem abexex

Step	Hyp	Ref	Expression
1		df-rex 2526	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		abexex.2	. . . . . 6 |- ( ph -> x e. A )
3	2	pm4.71ri 392	. . . . 5 |- ( ph <-> ( x e. A /\ ph ) )
4	3	exbii 1654	. . . 4 |- ( E. x ph <-> E. x ( x e. A /\ ph ) )
5	1, 4	bitr4i 187	. . 3 |- ( E. x e. A ph <-> E. x ph )
6	5	abbii 2348	. 2 |- { y | E. x e. A ph } = { y | E. x ph }
7		abexex.1	. . 3 |- A e. _V
8		abexex.3	. . 3 |- { y | ph } e. _V
9	7, 8	abrexex2 6317	. 2 |- { y | E. x e. A ph } e. _V
10	6, 9	eqeltrri 2306	1 |- { y | E. x ph } e. _V

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1541   e. wcel 2203   {cab 2218   E.wrex 2521   _Vcvv 2813

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by: (None)