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Theorem vtocleg 3553
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 2821 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 1928 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 17 1 (𝐴𝑉𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wex 1776  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-clel 2814
This theorem is referenced by:  vtoclg  3554  vtocleOLD  3556  spsbc  3804  snexg  5441  prex  5443  avril1  30492  bj-snexg  37017  rdgssun  37361  finxpreclem6  37379  ralssiun  37390  frege58c  43911  tz6.12i-afv2  47193
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