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Theorem bj-vtoclg1fv 32104
 Description: Version of bj-vtoclg1f 32103 with a dv condition on 𝑥, 𝑉. This removes dependency on df-sb 1868 and df-clab 2597. Prefer its use over bj-vtoclg1f 32103 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-vtoclg1fv.nf 𝑥𝜓
bj-vtoclg1fv.maj (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclg1fv.min 𝜑
Assertion
Ref Expression
bj-vtoclg1fv (𝐴𝑉𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-vtoclg1fv
StepHypRef Expression
1 bj-elissetv 32055 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-vtoclg1fv.nf . . 3 𝑥𝜓
3 bj-vtoclg1fv.maj . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 bj-vtoclg1fv.min . . 3 𝜑
52, 3, 4bj-exlimmpi 32097 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
61, 5syl 17 1 (𝐴𝑉𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-clel 2606 This theorem is referenced by: (None)
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