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Theorem vtocleg 3530
 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 3453 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 1931 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 17 1 (𝐴𝑉𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781   ∈ 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-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870 This theorem is referenced by:  vtocle  3533  spsbc  3735  prex  5302  avril1  28292  rdgssun  34946  finxpreclem6  34964  ralssiun  34975  frege58c  40793  tz6.12i-afv2  43967
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