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Theorem axac3 10502
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10501 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
Assertion
Ref Expression
axac3 CHOICE

Proof of Theorem axac3
Dummy variables 𝑤 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-ac2 10501 . . 3 𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
21ax-gen 1792 . 2 𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
3 dfackm 10205 . 2 (CHOICE ↔ ∀𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤))))))
42, 3mpbir 231 1 CHOICE
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  wal 1535  wex 1776  CHOICEwac 10153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ac 10154
This theorem is referenced by:  ackm  10503  axac  10505  axaci  10506  cardeqv  10507  fin71ac  10571  lbsex  21185  ptcls  23640  ptcmp  24082  axac10  43022
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