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Theorem axac3 9608
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 9607 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 9607 . . 3 𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
21ax-gen 1894 . 2 𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
3 dfackm 9310 . 2 (CHOICE ↔ ∀𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤))))))
42, 3mpbir 223 1 CHOICE
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 878  wal 1654  wex 1878  CHOICEwac 9258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-ac2 9607
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ac 9259
This theorem is referenced by:  ackm  9609  axac  9611  axaci  9612  cardeqv  9613  fin71ac  9677  lbsex  19533  ptcls  21797  ptcmp  22239  axac10  38438
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