Theorem abrexex2 3871

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 3860.

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-17 971	. . . 4 |- (E.x e. A ph -> A.zE.x e. A ph)
2		ax-17 971	. . . . 5 |- (x e. A -> A.y x e. A)
3		hbs1 1332	. . . . 5 |- ([z / y]ph -> A.y[z / y]ph)
4	2, 3	hbrex 1688	. . . 4 |- (E.x e. A [z / y]ph -> A.yE.x e. A [z / y]ph)
5		sbequ12 1181	. . . . 5 |- (y = z -> (ph <-> [z / y]ph))
6	5	rexbidv 1664	. . . 4 |- (y = z -> (E.x e. A ph <-> E.x e. A [z / y]ph))
7	1, 4, 6	cbvab 1908	. . 3 |- {y | E.x e. A ph} = {z | E.x e. A [z / y]ph}
8		df-clab 1464	. . . . 5 |- (z e. {y | ph} <-> [z / y]ph)
9	8	rexbii 1668	. . . 4 |- (E.x e. A z e. {y | ph} <-> E.x e. A [z / y]ph)
10	9	abbii 1575	. . 3 |- {z | E.x e. A z e. {y | ph}} = {z | E.x e. A [z / y]ph}
11	7, 10	eqtr4 1498	. 2 |- {y | E.x e. A ph} = {z | E.x e. A z e. {y | ph}}
12		df-iun 2568	. . 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 3863	. . 3 |- U_x e. A {y | ph} e. V
16	12, 15	eqeltrr 1545	. 2 |- {z | E.x e. A z e. {y | ph}} e. V
17	11, 16	eqeltr 1544	1 |- {y | E.x e. A ph} e. V

Syntax hints:   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  E.wrex 1646  Vcvv 1811  U_ciun 2566

This theorem is referenced by:  abexssex 3872  abexex 3873  brdom7disj 4804  brdom6disj 4805  sumex 6981  infxpidmlem9 7560

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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