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Theorem vtocldf 2995
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 424 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ps  <->  ch )
) )
63, 5alrimi 1781 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ps 
<->  ch ) ) )
7 vtocld.3 . . 3  |-  ( ph  ->  ps )
83, 7alrimi 1781 . 2  |-  ( ph  ->  A. x ps )
9 vtocld.1 . 2  |-  ( ph  ->  A  e.  V )
10 vtoclgft 2994 . 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 1195 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558
This theorem is referenced by:  vtocld  2996  iota2df  5434  riotasv2d  6586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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