Theorem abexssex 3857

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 1642	. . . 4 |- (E.x e. P~ Aph <-> E.x(x e. P~A /\ ph))
2		visset 1804	. . . . . . 7 |- x e. V
3	2	elpw 2394	. . . . . 6 |- (x e. P~A <-> x (_ A)
4	3	anbi1i 480	. . . . 5 |- ((x e. P~A /\ ph) <-> (x (_ A /\ ph))
5	4	exbii 1047	. . . 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 1567	. 2 |- {y | E.x e. P~ Aph} = {y | E.x(x (_ A /\ ph)}
8		abrexex2.1	. . . 4 |- A e. V
9	8	pwex 2735	. . 3 |- P~A e. V
10		abrexex2.2	. . 3 |- {y | ph} e. V
11	9, 10	abrexex2 3856	. 2 |- {y | E.x e. P~ Aph} e. V
12	7, 11	eqeltrr 1537	1 |- {y | E.x(x (_ A /\ ph)} e. V

Syntax hints:   /\ wa 223   e. wcel 955  E.wex 977  {cab 1456  E.wrex 1638  Vcvv 1802   (_ wss 2037  P~cpw 2391

This theorem is referenced by:  abfii2 4536  abfii4 4538  subbas 7586

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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