Theorem abexex 6094

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 2450	. . . 4 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		abexex.2	. . . . . 6 |- ( ph -> x e. A )
3	2	pm4.71ri 390	. . . . 5 |- ( ph <-> ( x e. A /\ ph ) )
4	3	exbii 1593	. . . 4 |- ( E. x ph <-> E. x ( x e. A /\ ph ) )
5	1, 4	bitr4i 186	. . 3 |- ( E. x e. A ph <-> E. x ph )
6	5	abbii 2282	. 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 6092	. 2 |- { y | E. x e. A ph } e. _V
10	6, 9	eqeltrri 2240	1 |- { y | E. x ph } e. _V

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   {cab 2151   E.wrex 2445   _Vcvv 2726

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by: (None)