Theorem abrexex2 3856

Description: Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. This is a powerful generalization of abrexex 3845.

Hypotheses

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

Assertion

Ref	Expression
abrexex2	|- {y | E.x e. A ph} e. V

Distinct variable group:   x,y,A

Proof of Theorem abrexex2

Step	Hyp	Ref	Expression
1		ax-17 968	. . . 4 |- (E.x e. A ph -> A.zE.x e. A ph)
2		ax-17 968	. . . . 5 |- (x e. A -> A.y x e. A)
3		hbs1 1327	. . . . 5 |- ([z / y]ph -> A.y[z / y]ph)
4	2, 3	hbrex 1680	. . . 4 |- (E.x e. A [z / y]ph -> A.yE.x e. A [z / y]ph)
5		sbequ12 1177	. . . . 5 |- (y = z -> (ph <-> [z / y]ph))
6	5	rexbidv 1656	. . . 4 |- (y = z -> (E.x e. A ph <-> E.x e. A [z / y]ph))
7	1, 4, 6	cbvab 1899	. . 3 |- {y | E.x e. A ph} = {z | E.x e. A [z / y]ph}
8		df-clab 1457	. . . . 5 |- (z e. {y | ph} <-> [z / y]ph)
9	8	rexbii 1660	. . . 4 |- (E.x e. A z e. {y | ph} <-> E.x e. A [z / y]ph)
10	9	abbii 1567	. . 3 |- {z | E.x e. A z e. {y | ph}} = {z | E.x e. A [z / y]ph}
11	7, 10	eqtr4 1490	. 2 |- {y | E.x e. A ph} = {z | E.x e. A z e. {y | ph}}
12		df-iun 2558	. . 3 |- U_x e. A {y | ph} = {z | E.x e. A z e. {y | ph}}
13		abrexex2.1	. . . 4 |- A e. V
14		abrexex2.2	. . . 4 |- {y | ph} e. V
15	13, 14	iunex 3848	. . 3 |- U_x e. A {y | ph} e. V
16	12, 15	eqeltrr 1537	. 2 |- {z | E.x e. A z e. {y | ph}} e. V
17	11, 16	eqeltr 1536	1 |- {y | E.x e. A ph} e. V

Syntax hints:   = wceq 953   e. wcel 955  [wsbc 1166  {cab 1456  E.wrex 1638  Vcvv 1802  U_ciun 2556

This theorem is referenced by:  abexssex 3857  abexex 3858  brdom7disj 4776  brdom6disj 4777  sumex 6919  infxpidmlem9 7503

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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