Theorem abexssex 3872

Description: Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in ph.

Hypotheses

Ref	Expression
abrexex2.1	|- A e. V
abrexex2.2	|- {y | ph} e. V

Assertion

Ref	Expression
abexssex	|- {y | E.x(x (_ A /\ ph)} e. V

Distinct variable group:   x,y,A

Proof of Theorem abexssex

Step	Hyp	Ref	Expression
1		df-rex 1650	. . . 4 |- (E.x e. P~ Aph <-> E.x(x e. P~A /\ ph))
2		visset 1813	. . . . . . 7 |- x e. V
3	2	elpw 2404	. . . . . 6 |- (x e. P~A <-> x (_ A)
4	3	anbi1i 481	. . . . 5 |- ((x e. P~A /\ ph) <-> (x (_ A /\ ph))
5	4	exbii 1051	. . . 4 |- (E.x(x e. P~A /\ ph) <-> E.x(x (_ A /\ ph))
6	1, 5	bitr 173	. . 3 |- (E.x e. P~ Aph <-> E.x(x (_ A /\ ph))
7	6	abbii 1575	. 2 |- {y | E.x e. P~ Aph} = {y | E.x(x (_ A /\ ph)}
8		abrexex2.1	. . . 4 |- A e. V
9	8	pwex 2745	. . 3 |- P~A e. V
10		abrexex2.2	. . 3 |- {y | ph} e. V
11	9, 10	abrexex2 3871	. 2 |- {y | E.x e. P~ Aph} e. V
12	7, 11	eqeltrr 1545	1 |- {y | E.x(x (_ A /\ ph)} e. V

Syntax hints:   /\ wa 223   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  Vcvv 1811   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  abfii2OLD 4562  abfii4OLD 4564  subbasOLD 7644

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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