Theorem vtocl3 1844

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 1815	. . . 4 |- E.x x = A
3		vtocl3.2	. . . . 5 |- B e. V
4	3	isseti 1815	. . . 4 |- E.y y = B
5		vtocl3.3	. . . . 5 |- C e. V
6	5	isseti 1815	. . . 4 |- E.z z = C
7		eeeanv 1324	. . . . 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 1040	. . . . . 6 |- (E.z(x = A /\ y = B /\ z = C) -> E.z(ph -> ps))
11	10	19.22i2 1041	. . . . 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 916	. . 3 |- E.xE.yE.z(ph -> ps)
14		19.36v 1300	. . . . . . 7 |- (E.z(ph -> ps) <-> (A.zph -> ps))
15	14	exbii 1051	. . . . . 6 |- (E.yE.z(ph -> ps) <-> E.y(A.zph -> ps))
16		19.36v 1300	. . . . . 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 1051	. . . 4 |- (E.xE.yE.z(ph -> ps) <-> E.x(A.yA.zph -> ps))
19		19.36v 1300	. . . 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 983	. 2 |- A.yA.zph
24	21, 23	mpg 986	1 |- ps

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  caoprass 4054  caoprdistr 4059  ertr 4274

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-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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