MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl2 Structured version   Unicode version

Theorem vtocl2 3037
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl2.1  |-  A  e. 
_V
vtocl2.2  |-  B  e. 
_V
vtocl2.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
vtocl2.4  |-  ph
Assertion
Ref Expression
vtocl2  |-  ps
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem vtocl2
StepHypRef Expression
1 vtocl2.1 . . . . . 6  |-  A  e. 
_V
21isseti 2990 . . . . 5  |-  E. x  x  =  A
3 vtocl2.2 . . . . . 6  |-  B  e. 
_V
43isseti 2990 . . . . 5  |-  E. y 
y  =  B
5 eeanv 1932 . . . . . 6  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  <->  ( E. x  x  =  A  /\  E. y 
y  =  B ) )
6 vtocl2.3 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
76biimpd 207 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  ->  ps ) )
872eximi 1626 . . . . . 6  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
95, 8sylbir 213 . . . . 5  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
102, 4, 9mp2an 672 . . . 4  |-  E. x E. y ( ph  ->  ps )
11 19.36v 1915 . . . . 5  |-  ( E. y ( ph  ->  ps )  <->  ( A. y ph  ->  ps ) )
1211exbii 1634 . . . 4  |-  ( E. x E. y (
ph  ->  ps )  <->  E. x
( A. y ph  ->  ps ) )
1310, 12mpbi 208 . . 3  |-  E. x
( A. y ph  ->  ps )
141319.36aiv 1916 . 2  |-  ( A. x A. y ph  ->  ps )
15 vtocl2.4 . . 3  |-  ph
1615ax-gen 1591 . 2  |-  A. y ph
1714, 16mpg 1593 1  |-  ps
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2986
This theorem is referenced by:  caovord  6286  sornom  8458  wloglei  9884  ipodrsima  15347  mpfind  17634  monotoddzzfi  29295
  Copyright terms: Public domain W3C validator