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Theorem vtoclefex 33492
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 1871 . 2 𝑥(𝑥 = 𝐴𝜑)
4 vtoclegft 3420 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
51, 3, 4mp3an23 1565 1 (𝐴𝑉𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630   = wceq 1632  wnf 1857  wcel 2139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342
This theorem is referenced by:  finxpreclem2  33538
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