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Theorem bj-vtoclg1f1 31902
Description: The FOL content of vtoclg1f 3234 (hence not using ax-ext 2586, df-cleq 2599, df-nfc 2736, df-v 3171). Note the weakened "major" hypothesis and the dv condition between 𝑥 and 𝐴 (needed since the class-form non-free predicate is not available without ax-ext 2586; as a byproduct, this dispenses with ax-11 2020 and ax-13 2229). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-vtoclg1f1.nf 𝑥𝜓
bj-vtoclg1f1.maj (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclg1f1.min 𝜑
Assertion
Ref Expression
bj-vtoclg1f1 (∃𝑦 𝑦 = 𝐴𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-vtoclg1f1
StepHypRef Expression
1 bj-denotes 31852 . 2 (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
2 bj-vtoclg1f1.nf . . 3 𝑥𝜓
3 bj-vtoclg1f1.maj . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 bj-vtoclg1f1.min . . 3 𝜑
52, 3, 4bj-exlimmpi 31897 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
61, 5sylbi 205 1 (∃𝑦 𝑦 = 𝐴𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wex 1694  wnf 1698
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-sb 1867  df-clab 2593  df-clel 2602
This theorem is referenced by: (None)
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