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Theorem vtocld 3556
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (𝜑𝐴𝑉)
vtocld.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
vtocld.3 (𝜑𝜓)
Assertion
Ref Expression
vtocld (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtocld
StepHypRef Expression
1 vtocld.1 . 2 (𝜑𝐴𝑉)
2 vtocld.2 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
3 vtocld.3 . 2 (𝜑𝜓)
4 nfv 1911 . 2 𝑥𝜑
5 nfcvd 2978 . 2 (𝜑𝑥𝐴)
6 nfvd 1912 . 2 (𝜑 → Ⅎ𝑥𝜒)
71, 2, 3, 4, 5, 6vtocldf 3555 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963
This theorem is referenced by:  vtocl2d  3557  lmatfval  31079  lmatcl  31081  dvgrat  40644  dfatbrafv2b  43445  fnbrafv2b  43448
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