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Theorem vtocleg 2678
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)
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 2622 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 1530 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 14 1 (𝐴𝑉𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wex 1422  wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612
This theorem is referenced by:  vtocle  2681  spsbc  2835  prexg  3994  funimaexglem  5033  eloprabga  5643  bj-prexg  10987
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