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Theorem bj-vtoclg1f 31902
Description: Reprove vtoclg1f 3233 from bj-vtoclg1f1 31901. This removes dependency on ax-ext 2585, df-cleq 2598 and df-v 3170. Use bj-vtoclg1fv 31903 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-vtoclg1f.nf 𝑥𝜓
bj-vtoclg1f.maj (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclg1f.min 𝜑
Assertion
Ref Expression
bj-vtoclg1f (𝐴𝑉𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem bj-vtoclg1f
StepHypRef Expression
1 bj-elisset 31855 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-vtoclg1f.nf . . 3 𝑥𝜓
3 bj-vtoclg1f.maj . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 bj-vtoclg1f.min . . 3 𝜑
52, 3, 4bj-exlimmpi 31896 . 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 1975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-clel 2601
This theorem is referenced by: (None)
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