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Theorem 2bornot2b 27662
 Description: The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
2bornot2b (2 · 𝐵 ∨ ¬ 2 · 𝐵)

Proof of Theorem 2bornot2b
StepHypRef Expression
1 ax-1 6 . . 3 (¬ 2 · 𝐵 → (2 · 𝐵 → ¬ 2 · 𝐵))
2 ax-1 6 . . 3 (¬ 2 · 𝐵 → ((2 · 𝐵 → ¬ 2 · 𝐵) → ¬ 2 · 𝐵))
31, 2mpd 15 . 2 (¬ 2 · 𝐵 → ¬ 2 · 𝐵)
4 df-or 837 . 2 ((2 · 𝐵 ∨ ¬ 2 · 𝐵) ↔ (¬ 2 · 𝐵 → ¬ 2 · 𝐵))
53, 4mpbir 221 1 (2 · 𝐵 ∨ ¬ 2 · 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 836   class class class wbr 4786   · cmul 10143  2c2 11272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-or 837 This theorem is referenced by: (None)
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