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Theorem vtoclef 3312
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
vtoclef.1 𝑥𝜑
vtoclef.2 𝐴 ∈ V
vtoclef.3 (𝑥 = 𝐴𝜑)
Assertion
Ref Expression
vtoclef 𝜑
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclef
StepHypRef Expression
1 vtoclef.2 . . 3 𝐴 ∈ V
21isseti 3240 . 2 𝑥 𝑥 = 𝐴
3 vtoclef.1 . . 3 𝑥𝜑
4 vtoclef.3 . . 3 (𝑥 = 𝐴𝜑)
53, 4exlimi 2124 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
62, 5ax-mp 5 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wex 1744  wnf 1748  wcel 2030  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  nn0ind-raph  11515  finxpreclem2  33357
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