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Theorem vtoclg1f 2748
Description: Version of vtoclgf 2747 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1485 and ax-13 1492. (Contributed by BJ, 1-May-2019.)
Hypotheses
Ref Expression
vtoclg1f.nf 𝑥𝜓
vtoclg1f.maj (𝑥 = 𝐴 → (𝜑𝜓))
vtoclg1f.min 𝜑
Assertion
Ref Expression
vtoclg1f (𝐴𝑉𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclg1f
StepHypRef Expression
1 elex 2700 . 2 (𝐴𝑉𝐴 ∈ V)
2 isset 2695 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3 vtoclg1f.nf . . . 4 𝑥𝜓
4 vtoclg1f.min . . . . 5 𝜑
5 vtoclg1f.maj . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5mpbii 147 . . . 4 (𝑥 = 𝐴𝜓)
73, 6exlimi 1574 . . 3 (∃𝑥 𝑥 = 𝐴𝜓)
82, 7sylbi 120 . 2 (𝐴 ∈ V → 𝜓)
91, 8syl 14 1 (𝐴𝑉𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wnf 1437  wex 1469  wcel 1481  Vcvv 2689
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691
This theorem is referenced by:  opeliunxp2f  6142  summodclem2a  11181
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