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Theorem bj-vtoclg1fv 31904
Description: Version of bj-vtoclg1f 31903 with a dv condition on 𝑥, 𝑉. This removes dependency on df-sb 1867 and df-clab 2593. Prefer its use over bj-vtoclg1f 31903 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 31855 . 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 31897 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
61, 5syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wex 1694  wnf 1698  wcel 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700  df-clel 2602
This theorem is referenced by: (None)
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