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Theorem vtocld 2665
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1  |-  ( ph  ->  A  e.  V )
vtocld.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
vtocld.3  |-  ( ph  ->  ps )
Assertion
Ref Expression
vtocld  |-  ( ph  ->  ch )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    V( x)

Proof of Theorem vtocld
StepHypRef Expression
1 vtocld.1 . 2  |-  ( ph  ->  A  e.  V )
2 vtocld.2 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
3 vtocld.3 . 2  |-  ( ph  ->  ps )
4 nfv 1464 . 2  |-  F/ x ph
5 nfcvd 2226 . 2  |-  ( ph  -> 
F/_ x A )
6 nfvd 1465 . 2  |-  ( ph  ->  F/ x ch )
71, 2, 3, 4, 5, 6vtocldf 2664 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617
This theorem is referenced by:  funfvima3  5483  isbth  6613  frec2uzuzd  9730
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