Theorem abexex 6129

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 2461	. . . 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 1605	. . . 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 2293	. 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 6127	. 2 |- { y | E. x e. A ph } e. _V
10	6, 9	eqeltrri 2251	1 |- { y | E. x ph } e. _V

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1492   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2739

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226

This theorem is referenced by: (None)