Theorem vtocl3 2342

Description: Implicit substitution of classes for set variables. (The proof was shortened by Andrew Salmon, 8-Jun-2011.)

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	. . . . . . 7 |- A e. _V
2	1	isseti 2297	. . . . . 6 |- E.x x = A
3		vtocl3.2	. . . . . . 7 |- B e. _V
4	3	isseti 2297	. . . . . 6 |- E.y y = B
5		vtocl3.3	. . . . . . 7 |- C e. _V
6	5	isseti 2297	. . . . . 6 |- E.z z = C
7		eeeanv 1708	. . . . . . 7 |- (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	. . . . . . . . . 10 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
9	8	biimpd 170	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (ph -> ps))
10	9	eximi 1387	. . . . . . . 8 |- (E.z(x = A /\ y = B /\ z = C) -> E.z(ph -> ps))
11	10	2eximi 1388	. . . . . . 7 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) -> E.xE.yE.z(ph -> ps))
12	7, 11	sylbir 218	. . . . . 6 |- ((E.x x = A /\ E.y y = B /\ E.z z = C) -> E.xE.yE.z(ph -> ps))
13	2, 4, 6, 12	mp3an 1191	. . . . 5 |- E.xE.yE.z(ph -> ps)
14		19.36v 1679	. . . . . 6 |- (E.z(ph -> ps) <-> (A.zph -> ps))
15	14	2exbii 1399	. . . . 5 |- (E.xE.yE.z(ph -> ps) <-> E.xE.y(A.zph -> ps))
16	13, 15	mpbi 206	. . . 4 |- E.xE.y(A.zph -> ps)
17		19.36v 1679	. . . . 5 |- (E.y(A.zph -> ps) <-> (A.yA.zph -> ps))
18	17	exbii 1398	. . . 4 |- (E.xE.y(A.zph -> ps) <-> E.x(A.yA.zph -> ps))
19	16, 18	mpbi 206	. . 3 |- E.x(A.yA.zph -> ps)
20	19	19.36aiv 1680	. 2 |- (A.xA.yA.zph -> ps)
21		vtocl3.5	. . 3 |- ph
22	21	gen2 1329	. 2 |- A.yA.zph
23	20, 22	mpg 1332	1 |- ps

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  caoprass 4987  caoprdistr 4992  ertr 5332  ertr2 16257

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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