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Theorem vtocldf 2837
Description: Implicit substitution of a class for a set 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 423 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1747 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 vtocld.3 . . 3  |-  ( ph  ->  ps )
83, 7alrimi 1747 . 2  |-  ( ph  ->  A. x ps )
9 vtocld.1 . 2  |-  ( ph  ->  A  e.  V )
10 vtoclgft 2836 . 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 1193 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1529   F/wnf 1533    = wceq 1625    e. wcel 1686   F/_wnfc 2408
This theorem is referenced by:  vtocld  2838  riotasv2d  6351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792
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