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Theorem vtocl2 2674
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 2627 . . . . 5  |-  E. x  x  =  A
3 vtocl2.2 . . . . . 6  |-  B  e. 
_V
43isseti 2627 . . . . 5  |-  E. y 
y  =  B
5 eeanv 1855 . . . . . 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 142 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  ->  ps ) )
872eximi 1537 . . . . . 6  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
95, 8sylbir 133 . . . . 5  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
102, 4, 9mp2an 417 . . . 4  |-  E. x E. y ( ph  ->  ps )
11 nfv 1466 . . . . 5  |-  F/ y ps
121119.36-1 1608 . . . 4  |-  ( E. y ( ph  ->  ps )  ->  ( A. y ph  ->  ps )
)
1310, 12eximii 1538 . . 3  |-  E. x
( A. y ph  ->  ps )
141319.36aiv 1829 . 2  |-  ( A. x A. y ph  ->  ps )
15 vtocl2.4 . . 3  |-  ph
1615ax-gen 1383 . 2  |-  A. y ph
1714, 16mpg 1385 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  undifexmid  4026  caovord  5808
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