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Theorem vtocldf 2605
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 )
vtocldf.4  |-  F/ x ph
vtocldf.5  |-  ( ph  -> 
F/_ x A )
vtocldf.6  |-  ( ph  ->  F/ x ch )
Assertion
Ref Expression
vtocldf  |-  ( ph  ->  ch )

Proof of Theorem vtocldf
StepHypRef Expression
1 vtocldf.5 . 2  |-  ( ph  -> 
F/_ x A )
2 vtocldf.6 . 2  |-  ( ph  ->  F/ x ch )
3 vtocldf.4 . . 3  |-  F/ x ph
4 vtocld.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
54ex 108 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1415 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 vtocld.3 . . 3  |-  ( ph  ->  ps )
83, 7alrimi 1415 . 2  |-  ( ph  ->  A. x ps )
9 vtocld.1 . 2  |-  ( ph  ->  A  e.  V )
10 vtoclgft 2604 . 2  |-  ( ( ( F/_ x A  /\  F/ x ch )  /\  ( A. x ( x  =  A  ->  ( ps  <->  ch ) )  /\  A. x ps )  /\  A  e.  V )  ->  ch )
111, 2, 6, 8, 9, 10syl221anc 1146 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243   F/wnf 1349    e. wcel 1393   F/_wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  vtocld  2606  peano2  4318  iota2df  4891
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