Theorem vtocl3 2610

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 2563	. . . . . 6 |- E. x x = A
3		vtocl3.2	. . . . . . 7 |- B e. _V
4	3	isseti 2563	. . . . . 6 |- E. y y = B
5		vtocl3.3	. . . . . . 7 |- C e. _V
6	5	isseti 2563	. . . . . 6 |- E. z z = C
7		eeeanv 1808	. . . . . . 7 |- ( E. x E. y E. 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 132	. . . . . . . . 9 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph -> ps ) )
10	9	eximi 1491	. . . . . . . 8 |- ( E. z ( x = A /\ y = B /\ z = C ) -> E. z ( ph -> ps ) )
11	10	2eximi 1492	. . . . . . 7 |- ( E. x E. y E. z ( x = A /\ y = B /\ z = C ) -> E. x E. y E. z ( ph -> ps ) )
12	7, 11	sylbir 125	. . . . . 6 |- ( ( E. x x = A /\ E. y y = B /\ E. z z = C ) -> E. x E. y E. z ( ph -> ps ) )
13	2, 4, 6, 12	mp3an 1232	. . . . 5 |- E. x E. y E. z ( ph -> ps )
14		nfv 1421	. . . . . . 7 |- F/ z ps
15	14	19.36-1 1563	. . . . . 6 |- ( E. z ( ph -> ps ) -> ( A. z ph -> ps ) )
16	15	2eximi 1492	. . . . 5 |- ( E. x E. y E. z ( ph -> ps ) -> E. x E. y ( A. z ph -> ps ) )
17	13, 16	ax-mp 7	. . . 4 |- E. x E. y ( A. z ph -> ps )
18		nfv 1421	. . . . 5 |- F/ y ps
19	18	19.36-1 1563	. . . 4 |- ( E. y ( A. z ph -> ps ) -> ( A. y A. z ph -> ps ) )
20	17, 19	eximii 1493	. . 3 |- E. x ( A. y A. z ph -> ps )
21	20	19.36aiv 1781	. 2 |- ( A. x A. y A. z ph -> ps )
22		vtocl3.5	. . 3 |- ph
23	22	gen2 1339	. 2 |- A. y A. z ph
24	21, 23	mpg 1340	1 |- ps

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 885   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)