Theorem vtocl3 1835

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 groups:   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 1806	. . . 4 |- E.x x = A
3		vtocl3.2	. . . . 5 |- B e. V
4	3	isseti 1806	. . . 4 |- E.y y = B
5		vtocl3.3	. . . . 5 |- C e. V
6	5	isseti 1806	. . . 4 |- E.z z = C
7		eeeanv 1319	. . . . 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	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
9	8	biimpd 153	. . . . . . 7 |- ((x = A /\ y = B /\ z = C) -> (ph -> ps))
10	9	19.22i 1036	. . . . . 6 |- (E.z(x = A /\ y = B /\ z = C) -> E.z(ph -> ps))
11	10	19.22i2 1037	. . . . 5 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) -> E.xE.yE.z(ph -> ps))
12	7, 11	sylbir 201	. . . 4 |- ((E.x x = A /\ E.y y = B /\ E.z z = C) -> E.xE.yE.z(ph -> ps))
13	2, 4, 6, 12	mp3an 913	. . 3 |- E.xE.yE.z(ph -> ps)
14		19.36v 1295	. . . . . . 7 |- (E.z(ph -> ps) <-> (A.zph -> ps))
15	14	exbii 1047	. . . . . 6 |- (E.yE.z(ph -> ps) <-> E.y(A.zph -> ps))
16		19.36v 1295	. . . . . 6 |- (E.y(A.zph -> ps) <-> (A.yA.zph -> ps))
17	15, 16	bitr 173	. . . . 5 |- (E.yE.z(ph -> ps) <-> (A.yA.zph -> ps))
18	17	exbii 1047	. . . 4 |- (E.xE.yE.z(ph -> ps) <-> E.x(A.yA.zph -> ps))
19		19.36v 1295	. . . 4 |- (E.x(A.yA.zph -> ps) <-> (A.xA.yA.zph -> ps))
20	18, 19	bitr 173	. . 3 |- (E.xE.yE.z(ph -> ps) <-> (A.xA.yA.zph -> ps))
21	13, 20	mpbi 189	. 2 |- (A.xA.yA.zph -> ps)
22		vtocl3.5	. . 3 |- ph
23	22	gen2 980	. 2 |- A.yA.zph
24	21, 23	mpg 983	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 773  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802

This theorem is referenced by:  caoprass 4040  caoprdistr 4045  ertr 4258

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-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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