Theorem vtocl3 2834

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 2785	. . . . . 6 |- E. x x = A
3		vtocl3.2	. . . . . . 7 |- B e. _V
4	3	isseti 2785	. . . . . 6 |- E. y y = B
5		vtocl3.3	. . . . . . 7 |- C e. _V
6	5	isseti 2785	. . . . . 6 |- E. z z = C
7		eeeanv 1962	. . . . . . 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 144	. . . . . . . . 9 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph -> ps ) )
10	9	eximi 1624	. . . . . . . 8 |- ( E. z ( x = A /\ y = B /\ z = C ) -> E. z ( ph -> ps ) )
11	10	2eximi 1625	. . . . . . 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 135	. . . . . 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 1350	. . . . 5 |- E. x E. y E. z ( ph -> ps )
14		nfv 1552	. . . . . . 7 |- F/ z ps
15	14	19.36-1 1697	. . . . . 6 |- ( E. z ( ph -> ps ) -> ( A. z ph -> ps ) )
16	15	2eximi 1625	. . . . 5 |- ( E. x E. y E. z ( ph -> ps ) -> E. x E. y ( A. z ph -> ps ) )
17	13, 16	ax-mp 5	. . . 4 |- E. x E. y ( A. z ph -> ps )
18		nfv 1552	. . . . 5 |- F/ y ps
19	18	19.36-1 1697	. . . 4 |- ( E. y ( A. z ph -> ps ) -> ( A. y A. z ph -> ps ) )
20	17, 19	eximii 1626	. . 3 |- E. x ( A. y A. z ph -> ps )
21	20	19.36aiv 1926	. 2 |- ( A. x A. y A. z ph -> ps )
22		vtocl3.5	. . 3 |- ph
23	22	gen2 1474	. 2 |- A. y A. z ph
24	21, 23	mpg 1475	1 |- ps

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778

This theorem is referenced by: (None)