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Theorem acexmid 5890
Description: The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer] p. 483.

The statement of the axiom of choice given here is ac2 in the Metamath Proof Explorer (version of 3-Aug-2019). In particular, note that the choice function 𝑦 provides a value when 𝑧 is inhabited (as opposed to nonempty as in some statements of the axiom of choice).

Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967).

For this theorem stated using the df-ac 7223 and df-exmid 4210 syntaxes, see exmidac 7226. (Contributed by Jim Kingdon, 4-Aug-2019.)

Hypothesis
Ref Expression
acexmid.choice 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
Assertion
Ref Expression
acexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem acexmid
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1539 . . . . . . . . . . . . . 14 𝑣(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒))
21sb8eu 2051 . . . . . . . . . . . . 13 (∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣[𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)))
3 eleq12 2254 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 = 𝑣𝑐 = 𝑧) → (𝑓𝑐𝑣𝑧))
43ancoms 268 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝑧𝑓 = 𝑣) → (𝑓𝑐𝑣𝑧))
543adant3 1019 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → (𝑓𝑐𝑣𝑧))
6 eleq12 2254 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝑧𝑒 = 𝑢) → (𝑐𝑒𝑧𝑢))
763ad2antl1 1161 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑐𝑒𝑧𝑢))
8 eleq12 2254 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝑣𝑒 = 𝑢) → (𝑓𝑒𝑣𝑢))
983ad2antl2 1162 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑓𝑒𝑣𝑢))
107, 9anbi12d 473 . . . . . . . . . . . . . . . . . . 19 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → ((𝑐𝑒𝑓𝑒) ↔ (𝑧𝑢𝑣𝑢)))
11 simpl3 1004 . . . . . . . . . . . . . . . . . . 19 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → 𝑏 = 𝑦)
1210, 11cbvrexdva2 2726 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → (∃𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
135, 12anbi12d 473 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
14133com23 1211 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑧𝑏 = 𝑦𝑓 = 𝑣) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
15143expa 1205 . . . . . . . . . . . . . . 15 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑓 = 𝑣) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
1615sbiedv 1800 . . . . . . . . . . . . . 14 ((𝑐 = 𝑧𝑏 = 𝑦) → ([𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
1716eubidv 2046 . . . . . . . . . . . . 13 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑣[𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
182, 17bitrid 192 . . . . . . . . . . . 12 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
19 df-reu 2475 . . . . . . . . . . . 12 (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)))
20 df-reu 2475 . . . . . . . . . . . 12 (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2118, 19, 203bitr4g 223 . . . . . . . . . . 11 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2221adantr 276 . . . . . . . . . 10 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
23 simpll 527 . . . . . . . . . 10 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → 𝑐 = 𝑧)
2422, 23cbvraldva2 2725 . . . . . . . . 9 ((𝑐 = 𝑧𝑏 = 𝑦) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2524ancoms 268 . . . . . . . 8 ((𝑏 = 𝑦𝑐 = 𝑧) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2625adantll 476 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
27 simpll 527 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → 𝑎 = 𝑥)
2826, 27cbvraldva2 2725 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → (∀𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2928cbvexdva 1941 . . . . 5 (𝑎 = 𝑥 → (∃𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3029cbvalv 1929 . . . 4 (∀𝑎𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑥𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
31 acexmid.choice . . . 4 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
3230, 31mpgbir 1464 . . 3 𝑎𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒)
3332spi 1547 . 2 𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒)
3433acexmidlemv 5889 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wo 709  w3a 980  wal 1362  wex 1503  [wsb 1773  ∃!weu 2038  wral 2468  wrex 2469  ∃!wreu 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386  df-iota 5193  df-riota 5847
This theorem is referenced by: (None)
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