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Theorem vtocl2 2713
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 2666 . . . . 5  |-  E. x  x  =  A
3 vtocl2.2 . . . . . 6  |-  B  e. 
_V
43isseti 2666 . . . . 5  |-  E. y 
y  =  B
5 eeanv 1882 . . . . . 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 143 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  ->  ps ) )
872eximi 1563 . . . . . 6  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
95, 8sylbir 134 . . . . 5  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
102, 4, 9mp2an 420 . . . 4  |-  E. x E. y ( ph  ->  ps )
11 nfv 1491 . . . . 5  |-  F/ y ps
121119.36-1 1634 . . . 4  |-  ( E. y ( ph  ->  ps )  ->  ( A. y ph  ->  ps )
)
1310, 12eximii 1564 . . 3  |-  E. x
( A. y ph  ->  ps )
141319.36aiv 1855 . 2  |-  ( A. x A. y ph  ->  ps )
15 vtocl2.4 . . 3  |-  ph
1615ax-gen 1408 . 2  |-  A. y ph
1714, 16mpg 1410 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  undifexmid  4085  caovord  5908  ctssexmid  6990
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