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Theorem vtocleg 3513
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 2816 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 1934 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 17 1 (𝐴𝑉𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wex 1782  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-clel 2811
This theorem is referenced by:  vtoclg  3524  vtocle  3543  spsbc  3753  snexg  5388  prex  5390  avril1  29449  bj-snexg  35551  rdgssun  35895  finxpreclem6  35913  ralssiun  35924  frege58c  42281  tz6.12i-afv2  45561
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