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Theorem vtoclefex 35053
 Description: Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
Hypotheses
Ref Expression
vtoclefex.1 𝑥𝜑
vtoclefex.3 (𝑥 = 𝐴𝜑)
Assertion
Ref Expression
vtoclefex (𝐴𝑉𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclefex
StepHypRef Expression
1 vtoclefex.1 . 2 𝑥𝜑
2 vtoclefex.3 . . 3 (𝑥 = 𝐴𝜑)
32ax-gen 1797 . 2 𝑥(𝑥 = 𝐴𝜑)
4 vtoclegft 3500 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
51, 3, 4mp3an23 1450 1 (𝐴𝑉𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-clel 2830 This theorem is referenced by:  finxpreclem2  35109
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