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Theorem vtocleg 2690
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 2633 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 1534 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 14 1 (𝐴𝑉𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wex 1426  wcel 1438
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  vtocle  2693  spsbc  2851  prexg  4038  funimaexglem  5097  eloprabga  5735  bj-prexg  11757
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