Theorem abexex 3858

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 1642	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		abexex.2	. . . . . 6 |- (ph -> x e. A)
3	2	pm4.71ri 636	. . . . 5 |- (ph <-> (x e. A /\ ph))
4	3	exbii 1047	. . . 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 1567	. 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 3856	. 2 |- {y | E.x e. A ph} e. V
10	6, 9	eqeltrr 1537	1 |- {y | E.xph} e. V

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  E.wex 977  {cab 1456  E.wrex 1638  Vcvv 1802

This theorem is referenced by:  brdom7disj 4776  brdom6disj 4777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188

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