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Theorem vtocl3 2714
Description: Implicit substitution of classes for setvar 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 2666 . . . . . 6  |-  E. x  x  =  A
3 vtocl3.2 . . . . . . 7  |-  B  e. 
_V
43isseti 2666 . . . . . 6  |-  E. y 
y  =  B
5 vtocl3.3 . . . . . . 7  |-  C  e. 
_V
65isseti 2666 . . . . . 6  |-  E. z 
z  =  C
7 eeeanv 1883 . . . . . . 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 143 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  ->  ps ) )
109eximi 1562 . . . . . . . 8  |-  ( E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. z
( ph  ->  ps )
)
11102eximi 1563 . . . . . . 7  |-  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  E. x E. y E. z (
ph  ->  ps ) )
127, 11sylbir 134 . . . . . 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 1298 . . . . 5  |-  E. x E. y E. z (
ph  ->  ps )
14 nfv 1491 . . . . . . 7  |-  F/ z ps
151419.36-1 1634 . . . . . 6  |-  ( E. z ( ph  ->  ps )  ->  ( A. z ph  ->  ps )
)
16152eximi 1563 . . . . 5  |-  ( E. x E. y E. z ( ph  ->  ps )  ->  E. x E. y ( A. z ph  ->  ps ) )
1713, 16ax-mp 5 . . . 4  |-  E. x E. y ( A. z ph  ->  ps )
18 nfv 1491 . . . . 5  |-  F/ y ps
191819.36-1 1634 . . . 4  |-  ( E. y ( A. z ph  ->  ps )  -> 
( A. y A. z ph  ->  ps )
)
2017, 19eximii 1564 . . 3  |-  E. x
( A. y A. z ph  ->  ps )
212019.36aiv 1855 . 2  |-  ( A. x A. y A. z ph  ->  ps )
22 vtocl3.5 . . 3  |-  ph
2322gen2 1409 . 2  |-  A. y A. z ph
2421, 23mpg 1410 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    /\ w3a 945   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-3an 947  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by: (None)
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