Theorem 2bornot2b 10143

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.)

Assertion

Ref	Expression
2bornot2b	|- (2 x. B \/ -. 2 x. B)

Proof of Theorem 2bornot2b

Step	Hyp	Ref	Expression
1		ax-1 4	. . 3 |- (-. 2 x. B -> (2 x. B -> -. 2 x. B))
2		ax-1 4	. . 3 |- (-. 2 x. B -> ((2 x. B -> -. 2 x. B) -> -. 2 x. B))
3	1, 2	mpd 29	. 2 |- (-. 2 x. B -> -. 2 x. B)
4		df-or 241	. 2 |- ((2 x. B \/ -. 2 x. B) <-> (-. 2 x. B -> -. 2 x. B))
5	3, 4	mpbir 207	1 |- (2 x. B \/ -. 2 x. B)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   class class class wbr 3338   x. cmul 6391  2c2 7145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-or 241

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