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Theorem axac3 10504
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10503 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 10503 . . 3 𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
21ax-gen 1795 . 2 𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
3 dfackm 10207 . 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 848  wal 1538  wex 1779  CHOICEwac 10155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ac 10156
This theorem is referenced by:  ackm  10505  axac  10507  axaci  10508  cardeqv  10509  fin71ac  10573  lbsex  21167  ptcls  23624  ptcmp  24066  axac10  43045
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