MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axac3 Structured version   Visualization version   GIF version

Theorem axac3 10495
Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 10494 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 10494 . . 3 𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
21ax-gen 1789 . 2 𝑥𝑦𝑧𝑤𝑣((𝑦𝑥 ∧ (𝑧𝑦 → ((𝑤𝑥 ∧ ¬ 𝑦 = 𝑤) ∧ 𝑧𝑤))) ∨ (¬ 𝑦𝑥 ∧ (𝑧𝑥 → ((𝑤𝑧𝑤𝑦) ∧ ((𝑣𝑧𝑣𝑦) → 𝑣 = 𝑤)))))
3 dfackm 10197 . 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 394  wo 845  wal 1531  wex 1773  CHOICEwac 10146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-ac2 10494
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ac 10147
This theorem is referenced by:  ackm  10496  axac  10498  axaci  10499  cardeqv  10500  fin71ac  10564  lbsex  21060  ptcls  23540  ptcmp  23982  axac10  42485
  Copyright terms: Public domain W3C validator