Theorem abexex 3873

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 1650	. . . 4 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
2		abexex.2	. . . . . 6 |- (ph -> x e. A)
3	2	pm4.71ri 638	. . . . 5 |- (ph <-> (x e. A /\ ph))
4	3	exbii 1051	. . . 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 1575	. 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 3871	. 2 |- {y | E.x e. A ph} e. V
10	6, 9	eqeltrr 1545	1 |- {y | E.xph} e. V

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  Vcvv 1811

This theorem is referenced by:  brdom7disj 4804  brdom6disj 4805

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198

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