MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl3 Unicode version

Theorem vtocl3 2952
Description: Implicit substitution of classes for set 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
StepHypRef Expression
1 vtocl3.1 . . . . . . 7  |-  A  e. 
_V
21isseti 2906 . . . . . 6  |-  E. x  x  =  A
3 vtocl3.2 . . . . . . 7  |-  B  e. 
_V
43isseti 2906 . . . . . 6  |-  E. y 
y  =  B
5 vtocl3.3 . . . . . . 7  |-  C  e. 
_V
65isseti 2906 . . . . . 6  |-  E. z 
z  =  C
7 eeeanv 1927 . . . . . . 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 )
)
98biimpd 199 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  ->  ps ) )
109eximi 1582 . . . . . . . 8  |-  ( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z
( ph  ->  ps )
)
11102eximi 1583 . . . . . . 7  |-  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. x E. y E. z (
ph  ->  ps ) )
127, 11sylbir 205 . . . . . 6  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B  /\  E. z  z  =  C )  ->  E. x E. y E. z (
ph  ->  ps ) )
132, 4, 6, 12mp3an 1279 . . . . 5  |-  E. x E. y E. z (
ph  ->  ps )
14 19.36v 1908 . . . . . 6  |-  ( E. z ( ph  ->  ps )  <->  ( A. z ph  ->  ps ) )
15142exbii 1590 . . . . 5  |-  ( E. x E. y E. z ( ph  ->  ps )  <->  E. x E. y
( A. z ph  ->  ps ) )
1613, 15mpbi 200 . . . 4  |-  E. x E. y ( A. z ph  ->  ps )
17 19.36v 1908 . . . . 5  |-  ( E. y ( A. z ph  ->  ps )  <->  ( A. y A. z ph  ->  ps ) )
1817exbii 1589 . . . 4  |-  ( E. x E. y ( A. z ph  ->  ps )  <->  E. x ( A. y A. z ph  ->  ps ) )
1916, 18mpbi 200 . . 3  |-  E. x
( A. y A. z ph  ->  ps )
201919.36aiv 1909 . 2  |-  ( A. x A. y A. z ph  ->  ps )
21 vtocl3.5 . . 3  |-  ph
2221gen2 1553 . 2  |-  A. y A. z ph
2320, 22mpg 1554 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-v 2902
  Copyright terms: Public domain W3C validator