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Theorem vtocleg 2729
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 2672 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtocleg.1 . . 3 (𝑥 = 𝐴𝜑)
32exlimiv 1560 . 2 (∃𝑥 𝑥 = 𝐴𝜑)
41, 3syl 14 1 (𝐴𝑉𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wex 1451  wcel 1463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  vtocle  2732  spsbc  2891  prexg  4101  funimaexglem  5174  eloprabga  5824  bj-prexg  12943
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