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Theorem 2bornot2b 27290
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 385 . 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 383   class class class wbr 4644   · cmul 9926  2c2 11055
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 385
This theorem is referenced by: (None)
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