Theorem vtocl3 1380

Description: Implicit substitution of classes for set variables.

Hypotheses

Ref	Expression
vtocl3.1	|- A e. V
vtocl3.2	|- B e. V
vtocl3.3	|- C e. V
vtocl3.4	|- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
vtocl3.5	|- ph

Assertion

Ref	Expression
vtocl3	|- ps

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

Proof of Theorem vtocl3

Step	Hyp	Ref	Expression
1		vtocl3.1	. . . . 5 |- A e. V
2	1	isseti 1352	. . . 4 |- E.x x = A
3		vtocl3.2	. . . . 5 |- B e. V
4	3	isseti 1352	. . . 4 |- E.y y = B
5		vtocl3.3	. . . . 5 |- C e. V
6	5	isseti 1352	. . . 4 |- E.z z = C
7		eeeanv 981	. . . . 5 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) <-> (E.x x = A /\ E.y y = B /\ E.z z = C))
8		vtocl3.4	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
9	8	biimpd 135	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> (ph -> ps))
10	9	19.22i 723	. . . . . . 7 |- (E.z(x = A /\ y = B /\ z = C) -> E.z(ph -> ps))
11	10	19.22i 723	. . . . . 6 |- (E.yE.z(x = A /\ y = B /\ z = C) -> E.yE.z(ph -> ps))
12	11	19.22i 723	. . . . 5 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) -> E.xE.yE.z(ph -> ps))
13	7, 12	sylbir 176	. . . 4 |- ((E.x x = A /\ E.y y = B /\ E.z z = C) -> E.xE.yE.z(ph -> ps))
14	2, 4, 6, 13	mp3an 642	. . 3 |- E.xE.yE.z(ph -> ps)
15		19.36v 958	. . . . . . 7 |- (E.z(ph -> ps) <-> (A.zph -> ps))
16	15	biex 733	. . . . . 6 |- (E.yE.z(ph -> ps) <-> E.y(A.zph -> ps))
17		19.36v 958	. . . . . 6 |- (E.y(A.zph -> ps) <-> (A.yA.zph -> ps))
18	16, 17	bitr 151	. . . . 5 |- (E.yE.z(ph -> ps) <-> (A.yA.zph -> ps))
19	18	biex 733	. . . 4 |- (E.xE.yE.z(ph -> ps) <-> E.x(A.yA.zph -> ps))
20		19.36v 958	. . . 4 |- (E.x(A.yA.zph -> ps) <-> (A.xA.yA.zph -> ps))
21	19, 20	bitr 151	. . 3 |- (E.xE.yE.z(ph -> ps) <-> (A.xA.yA.zph -> ps))
22	14, 21	mpbi 164	. 2 |- (A.xA.yA.zph -> ps)
23		vtocl3.5	. . 3 |- ph
24	23	gen2 681	. 2 |- A.yA.zph
25	22, 24	mpg 684	1 |- ps

Syntax hints:   -> wi 2   <-> wb 127   /\ w3a 581  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  caoprass 3068  caoprdistr 3073  ertr 3211

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-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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