Theorem vtocl3 3262

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 3209	. . . . . 6 |- E. x x = A
3		vtocl3.2	. . . . . . 7 |- B e. _V
4	3	isseti 3209	. . . . . 6 |- E. y y = B
5		vtocl3.3	. . . . . . 7 |- C e. _V
6	5	isseti 3209	. . . . . 6 |- E. z z = C
7		eeeanv 2183	. . . . . . 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 219	. . . . . . . . 9 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph -> ps ) )
10	9	eximi 1762	. . . . . . . 8 |- ( E. z ( x = A /\ y = B /\ z = C ) -> E. z ( ph -> ps ) )
11	10	2eximi 1763	. . . . . . 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 225	. . . . . 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 1424	. . . . 5 |- E. x E. y E. z ( ph -> ps )
14		19.36v 1904	. . . . . 6 |- ( E. z ( ph -> ps ) <-> ( A. z ph -> ps ) )
15	14	2exbii 1775	. . . . 5 |- ( E. x E. y E. z ( ph -> ps ) <-> E. x E. y ( A. z ph -> ps ) )
16	13, 15	mpbi 220	. . . 4 |- E. x E. y ( A. z ph -> ps )
17		19.36v 1904	. . . . 5 |- ( E. y ( A. z ph -> ps ) <-> ( A. y A. z ph -> ps ) )
18	17	exbii 1774	. . . 4 |- ( E. x E. y ( A. z ph -> ps ) <-> E. x ( A. y A. z ph -> ps ) )
19	16, 18	mpbi 220	. . 3 |- E. x ( A. y A. z ph -> ps )
20	19	19.36iv 1905	. 2 |- ( A. x A. y A. z ph -> ps )
21		vtocl3.5	. . 3 |- ph
22	21	gen2 1723	. 2 |- A. y A. z ph
23	20, 22	mpg 1724	1 |- ps

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)