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Theorem bj-vtoclg 36388
Description: A version of vtoclg 3538 with an additional disjoint variable condition (which is removable if we allow use of df-clab 2705, see bj-vtoclg1f 36386), which requires fewer axioms (i.e., removes dependency on ax-6 1964, ax-7 2004, ax-9 2109, ax-12 2164, ax-ext 2698, df-clab 2705, df-cleq 2719, df-v 3471). (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 2809 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-vtoclg.maj . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 bj-vtoclg.min . . 3 𝜑
42, 3bj-exlimvmpi 36379 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
51, 4syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wex 1774  wcel 2099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-clel 2805
This theorem is referenced by:  bj-zfauscl  36392
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