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Theorem vtocld 3131
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 1755 . 2  |-  F/ x ph
5 nfcvd 2581 . 2  |-  ( ph  -> 
F/_ x A )
6 nfvd 1756 . 2  |-  ( ph  ->  F/ x ch )
71, 2, 3, 4, 5, 6vtocldf 3130 1  |-  ( ph  ->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082
This theorem is referenced by:  lmatfval  28649  lmatcl  28651  dvgrat  36632
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