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Theorem vtocleg 3430
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
vtocleg.1 (𝑥 = 𝐴𝜑)
Assertion
Ref Expression
vtocleg (𝐴𝑉𝜑)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem vtocleg
StepHypRef Expression
1 elisset 3366 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 2010 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 17 1 (𝐴𝑉𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wex 1852  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  vtocle  3433  spsbc  3600  prex  5037  avril1  27661  finxpreclem6  33570  frege58c  38741
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