Theorem vtocl3 2912

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof 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

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem vtocl3

Step	Hyp	Ref	Expression
1		vtocl3.1	. . . . . . 7 |- A e. _V
2	1	isseti 2866	. . . . . 6 |- E.x x = A
3		vtocl3.2	. . . . . . 7 |- B e. _V
4	3	isseti 2866	. . . . . 6 |- E.y y = B
5		vtocl3.3	. . . . . . 7 |- C e. _V
6	5	isseti 2866	. . . . . 6 |- E.z z = C
7		eeeanv 1914	. . . . . . 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 198	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (ph -> ps))
10	9	eximi 1576	. . . . . . . 8 |- (E.z(x = A /\ y = B /\ z = C) -> E.z(ph -> ps))
11	10	2eximi 1577	. . . . . . 7 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) -> E.xE.yE.z(ph -> ps))
12	7, 11	sylbir 204	. . . . . 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 1277	. . . . 5 |- E.xE.yE.z(ph -> ps)
14		19.36v 1896	. . . . . 6 |- (E.z(ph -> ps) <-> (A.zph -> ps))
15	14	2exbii 1583	. . . . 5 |- (E.xE.yE.z(ph -> ps) <-> E.xE.y(A.zph -> ps))
16	13, 15	mpbi 199	. . . 4 |- E.xE.y(A.zph -> ps)
17		19.36v 1896	. . . . 5 |- (E.y(A.zph -> ps) <-> (A.yA.zph -> ps))
18	17	exbii 1582	. . . 4 |- (E.xE.y(A.zph -> ps) <-> E.x(A.yA.zph -> ps))
19	16, 18	mpbi 199	. . 3 |- E.x(A.yA.zph -> ps)
20	19	19.36aiv 1897	. 2 |- (A.xA.yA.zph -> ps)
21		vtocl3.5	. . 3 |- ph
22	21	gen2 1547	. 2 |- A.yA.zph
23	20, 22	mpg 1548	1 |- ps

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)