Theorem abexex 3879

Description: A condition where a class builder continues to exist after its wff is existentially quantified.

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.xph} e. V

Distinct variable group:   x,y,A

Proof of Theorem abexex

Step	Hyp	Ref	Expression
1		df-rex 1653	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		abexex.2	. . . . . 6 |- (ph -> x e. A)
3	2	pm4.71ri 640	. . . . 5 |- (ph <-> (x e. A /\ ph))
4	3	exbii 1053	. . . 4 |- (E.xph <-> E.x(x e. A /\ ph))
5	1, 4	bitr4 176	. . 3 |- (E.x e. A ph <-> E.xph)
6	5	abbii 1578	. 2 |- {y | E.x e. A ph} = {y | E.xph}
7		abexex.1	. . 3 |- A e. V
8		abexex.3	. . 3 |- {y | ph} e. V
9	7, 8	abrexex2 3877	. 2 |- {y | E.x e. A ph} e. V
10	6, 9	eqeltrr 1548	1 |- {y | E.xph} e. V

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649  Vcvv 1814

This theorem is referenced by:  brdom7disj 4814  brdom6disj 4815

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

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