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Theorem vtoclb 3161
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)
Hypotheses
Ref Expression
vtoclb.1  |-  A  e. 
_V
vtoclb.2  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
vtoclb.3  |-  ( x  =  A  ->  ( ps 
<->  th ) )
vtoclb.4  |-  ( ph  <->  ps )
Assertion
Ref Expression
vtoclb  |-  ( ch  <->  th )
Distinct variable groups:    x, A    ch, x    th, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem vtoclb
StepHypRef Expression
1 vtoclb.1 . 2  |-  A  e. 
_V
2 vtoclb.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
3 vtoclb.3 . . 3  |-  ( x  =  A  ->  ( ps 
<->  th ) )
42, 3bibi12d 319 . 2  |-  ( x  =  A  ->  (
( ph  <->  ps )  <->  ( ch  <->  th ) ) )
5 vtoclb.4 . 2  |-  ( ph  <->  ps )
61, 4, 5vtocl 3158 1  |-  ( ch  <->  th )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   _Vcvv 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108
This theorem is referenced by:  alexeq  3226  sbss  3927  bnj609  34376
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