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Theorem axac3 10488
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10487 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 10487 . . 3 𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
21ax-gen 1790 . 2 𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
3 dfackm 10190 . 2 (CHOICE ↔ ∀𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤))))))
42, 3mpbir 230 1 CHOICE
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846  wal 1532  wex 1774  CHOICEwac 10139
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  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-ac2 10487
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ac 10140
This theorem is referenced by:  ackm  10489  axac  10491  axaci  10492  cardeqv  10493  fin71ac  10557  lbsex  21053  ptcls  23533  ptcmp  23975  axac10  42454
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