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Theorem vtocldf 2781
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 114 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1515 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 vtocld.3 . . 3  |-  ( ph  ->  ps )
83, 7alrimi 1515 . 2  |-  ( ph  ->  A. x ps )
9 vtocld.1 . 2  |-  ( ph  ->  A  e.  V )
10 vtoclgft 2780 . 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 1244 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346    = wceq 1348   F/wnf 1453    e. wcel 2141   F/_wnfc 2299
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  vtocld  2782  peano2  4577  iota2df  5182
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