Theorem cbvop 2473

Description: Change restricted bound variable to two restricted bound variables.

Hypothesis

Ref	Expression
cbvop.1	|- (x = <.y, z>. -> (ph <-> ps))

Assertion

Ref	Expression
cbvop	|- (E.x e. (A X. B)ph <-> E.y e. A E.z e. B ps)

Distinct variable group(s):   ph,y,z   ps,x   x,y,z,A   x,B,y,z

Proof of Theorem cbvop

Step	Hyp	Ref	Expression
1		anass 336	. . . . 5 |- (((E.yE.z x = <.y, z>. /\ x e. (A X. B)) /\ ph) <-> (E.yE.z x = <.y, z>. /\ (x e. (A X. B) /\ ph)))
2		elxp3 2460	. . . . . . 7 |- (x e. (A X. B) <-> E.yE.z(<.y, z>. = x /\ <.y, z>. e. (A X. B)))
3		eleq1 1149	. . . . . . . . . 10 |- (<.y, z>. = x -> (<.y, z>. e. (A X. B) <-> x e. (A X. B)))
4	3	pm5.32i 489	. . . . . . . . 9 |- ((<.y, z>. = x /\ <.y, z>. e. (A X. B)) <-> (<.y, z>. = x /\ x e. (A X. B)))
5		cleqcom 1103	. . . . . . . . . 10 |- (x = <.y, z>. <-> <.y, z>. = x)
6	5	anbi1i 368	. . . . . . . . 9 |- ((x = <.y, z>. /\ x e. (A X. B)) <-> (<.y, z>. = x /\ x e. (A X. B)))
7	4, 6	bitr4 154	. . . . . . . 8 |- ((<.y, z>. = x /\ <.y, z>. e. (A X. B)) <-> (x = <.y, z>. /\ x e. (A X. B)))
8	7	bi2ex 734	. . . . . . 7 |- (E.yE.z(<.y, z>. = x /\ <.y, z>. e. (A X. B)) <-> E.yE.z(x = <.y, z>. /\ x e. (A X. B)))
9		19.41vv 964	. . . . . . 7 |- (E.yE.z(x = <.y, z>. /\ x e. (A X. B)) <-> (E.yE.z x = <.y, z>. /\ x e. (A X. B)))
10	2, 8, 9	3bitr 155	. . . . . 6 |- (x e. (A X. B) <-> (E.yE.z x = <.y, z>. /\ x e. (A X. B)))
11	10	anbi1i 368	. . . . 5 |- ((x e. (A X. B) /\ ph) <-> ((E.yE.z x = <.y, z>. /\ x e. (A X. B)) /\ ph))
12		19.41vv 964	. . . . 5 |- (E.yE.z(x = <.y, z>. /\ (x e. (A X. B) /\ ph)) <-> (E.yE.z x = <.y, z>. /\ (x e. (A X. B) /\ ph)))
13	1, 11, 12	3bitr4 158	. . . 4 |- ((x e. (A X. B) /\ ph) <-> E.yE.z(x = <.y, z>. /\ (x e. (A X. B) /\ ph)))
14	13	biex 733	. . 3 |- (E.x(x e. (A X. B) /\ ph) <-> E.xE.yE.z(x = <.y, z>. /\ (x e. (A X. B) /\ ph)))
15		exrot3 777	. . 3 |- (E.xE.yE.z(x = <.y, z>. /\ (x e. (A X. B) /\ ph)) <-> E.yE.zE.x(x = <.y, z>. /\ (x e. (A X. B) /\ ph)))
16		opex 1893	. . . . 5 |- <.y, z>. e. V
17		eleq1 1149	. . . . . 6 |- (x = <.y, z>. -> (x e. (A X. B) <-> <.y, z>. e. (A X. B)))
18		cbvop.1	. . . . . 6 |- (x = <.y, z>. -> (ph <-> ps))
19	17, 18	anbi12d 476	. . . . 5 |- (x = <.y, z>. -> ((x e. (A X. B) /\ ph) <-> (<.y, z>. e. (A X. B) /\ ps)))
20	16, 19	ceqsexv 1371	. . . 4 |- (E.x(x = <.y, z>. /\ (x e. (A X. B) /\ ph)) <-> (<.y, z>. e. (A X. B) /\ ps))
21	20	bi2ex 734	. . 3 |- (E.yE.zE.x(x = <.y, z>. /\ (x e. (A X. B) /\ ph)) <-> E.yE.z(<.y, z>. e. (A X. B) /\ ps))
22	14, 15, 21	3bitr 155	. 2 |- (E.x(x e. (A X. B) /\ ph) <-> E.yE.z(<.y, z>. e. (A X. B) /\ ps))
23		df-rex 1206	. 2 |- (E.x e. (A X. B)ph <-> E.x(x e. (A X. B) /\ ph))
24		r2ex 1241	. . 3 |- (E.y e. A E.z e. B ps <-> E.yE.z((y e. A /\ z e. B) /\ ps))
25		visset 1350	. . . . . 6 |- z e. V
26	25	opelxp 2452	. . . . 5 |- (<.y, z>. e. (A X. B) <-> (y e. A /\ z e. B))
27	26	anbi1i 368	. . . 4 |- ((<.y, z>. e. (A X. B) /\ ps) <-> ((y e. A /\ z e. B) /\ ps))
28	27	bi2ex 734	. . 3 |- (E.yE.z(<.y, z>. e. (A X. B) /\ ps) <-> E.yE.z((y e. A /\ z e. B) /\ ps))
29	24, 28	bitr4 154	. 2 |- (E.y e. A E.z e. B ps <-> E.yE.z(<.y, z>. e. (A X. B) /\ ps))
30	22, 23, 29	3bitr4 158	1 |- (E.x e. (A X. B)ph <-> E.y e. A E.z e. B ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  E.wrex 1202  <.cop 1810   X. cxp 2408

This theorem is referenced by:  elrnoprab 3054  oprvalex 3055

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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