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Theorem bj-vtoclg 36307
Description: A version of vtoclg 3537 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2704, see bj-vtoclg1f 36305), which requires fewer axioms (i.e., removes dependency on ax-6 1963, ax-7 2003, ax-9 2108, ax-12 2163, ax-ext 2697, df-clab 2704, df-cleq 2718, df-v 3470). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-vtoclg.maj (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclg.min 𝜑
Assertion
Ref Expression
bj-vtoclg (𝐴𝑉𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-vtoclg
StepHypRef Expression
1 elissetv 2808 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-vtoclg.maj . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 bj-vtoclg.min . . 3 𝜑
42, 3bj-exlimvmpi 36298 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
51, 4syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wex 1773  wcel 2098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-clel 2804
This theorem is referenced by:  bj-zfauscl  36311
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