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Theorem vtocl2 2626
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 2580 . . . . 5  |-  E. x  x  =  A
3 vtocl2.2 . . . . . 6  |-  B  e. 
_V
43isseti 2580 . . . . 5  |-  E. y 
y  =  B
5 eeanv 1823 . . . . . 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 136 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  ->  ps ) )
872eximi 1508 . . . . . 6  |-  ( E. x E. y ( x  =  A  /\  y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
95, 8sylbir 129 . . . . 5  |-  ( ( E. x  x  =  A  /\  E. y 
y  =  B )  ->  E. x E. y
( ph  ->  ps )
)
102, 4, 9mp2an 410 . . . 4  |-  E. x E. y ( ph  ->  ps )
11 nfv 1437 . . . . 5  |-  F/ y ps
121119.36-1 1579 . . . 4  |-  ( E. y ( ph  ->  ps )  ->  ( A. y ph  ->  ps )
)
1310, 12eximii 1509 . . 3  |-  E. x
( A. y ph  ->  ps )
141319.36aiv 1797 . 2  |-  ( A. x A. y ph  ->  ps )
15 vtocl2.4 . . 3  |-  ph
1615ax-gen 1354 . 2  |-  A. y ph
1714, 16mpg 1356 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  caovord  5700
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