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Theorem acexmid 5766
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 7055 and df-exmid 4114 syntaxes, see exmidac 7058. (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 1508 . . . . . . . . . . . . . 14 𝑣(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒))
21sb8eu 2010 . . . . . . . . . . . . 13 (∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣[𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)))
3 eleq12 2202 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 = 𝑣𝑐 = 𝑧) → (𝑓𝑐𝑣𝑧))
43ancoms 266 . . . . . . . . . . . . . . . . . . 19 ((𝑐 = 𝑧𝑓 = 𝑣) → (𝑓𝑐𝑣𝑧))
543adant3 1001 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → (𝑓𝑐𝑣𝑧))
6 eleq12 2202 . . . . . . . . . . . . . . . . . . . . 21 ((𝑐 = 𝑧𝑒 = 𝑢) → (𝑐𝑒𝑧𝑢))
763ad2antl1 1143 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑐𝑒𝑧𝑢))
8 eleq12 2202 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 = 𝑣𝑒 = 𝑢) → (𝑓𝑒𝑣𝑢))
983ad2antl2 1144 . . . . . . . . . . . . . . . . . . . 20 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → (𝑓𝑒𝑣𝑢))
107, 9anbi12d 464 . . . . . . . . . . . . . . . . . . 19 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → ((𝑐𝑒𝑓𝑒) ↔ (𝑧𝑢𝑣𝑢)))
11 simpl3 986 . . . . . . . . . . . . . . . . . . 19 (((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) ∧ 𝑒 = 𝑢) → 𝑏 = 𝑦)
1210, 11cbvrexdva2 2657 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → (∃𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
135, 12anbi12d 464 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑧𝑓 = 𝑣𝑏 = 𝑦) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
14133com23 1187 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑧𝑏 = 𝑦𝑓 = 𝑣) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
15143expa 1181 . . . . . . . . . . . . . . 15 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑓 = 𝑣) → ((𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
1615sbiedv 1762 . . . . . . . . . . . . . 14 ((𝑐 = 𝑧𝑏 = 𝑦) → ([𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ (𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
1716eubidv 2005 . . . . . . . . . . . . 13 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑣[𝑣 / 𝑓](𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
182, 17syl5bb 191 . . . . . . . . . . . 12 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢))))
19 df-reu 2421 . . . . . . . . . . . 12 (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑓(𝑓𝑐 ∧ ∃𝑒𝑏 (𝑐𝑒𝑓𝑒)))
20 df-reu 2421 . . . . . . . . . . . 12 (∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃!𝑣(𝑣𝑧 ∧ ∃𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2118, 19, 203bitr4g 222 . . . . . . . . . . 11 ((𝑐 = 𝑧𝑏 = 𝑦) → (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2221adantr 274 . . . . . . . . . 10 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → (∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
23 simpll 518 . . . . . . . . . 10 (((𝑐 = 𝑧𝑏 = 𝑦) ∧ 𝑑 = 𝑤) → 𝑐 = 𝑧)
2422, 23cbvraldva2 2656 . . . . . . . . 9 ((𝑐 = 𝑧𝑏 = 𝑦) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2524ancoms 266 . . . . . . . 8 ((𝑏 = 𝑦𝑐 = 𝑧) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2625adantll 467 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → (∀𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
27 simpll 518 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑦) ∧ 𝑐 = 𝑧) → 𝑎 = 𝑥)
2826, 27cbvraldva2 2656 . . . . . 6 ((𝑎 = 𝑥𝑏 = 𝑦) → (∀𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
2928cbvexdva 1899 . . . . 5 (𝑎 = 𝑥 → (∃𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
3029cbvalv 1889 . . . 4 (∀𝑎𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒) ↔ ∀𝑥𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢))
31 acexmid.choice . . . 4 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
3230, 31mpgbir 1429 . . 3 𝑎𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒)
3332spi 1516 . 2 𝑏𝑐𝑎𝑑𝑐 ∃!𝑓𝑐𝑒𝑏 (𝑐𝑒𝑓𝑒)
3433acexmidlemv 5765 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wo 697  w3a 962  wal 1329  wex 1468  [wsb 1735  ∃!weu 1997  wral 2414  wrex 2415  ∃!wreu 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288  df-iota 5083  df-riota 5723
This theorem is referenced by: (None)
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