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Theorem 2bornot2b 21758
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  x.  B  \/  -.  2  x.  B )

Proof of Theorem 2bornot2b
StepHypRef Expression
1 ax-1 5 . . 3  |-  ( -.  2  x.  B  -> 
( 2  x.  B  ->  -.  2  x.  B
) )
2 ax-1 5 . . 3  |-  ( -.  2  x.  B  -> 
( ( 2  x.  B  ->  -.  2  x.  B )  ->  -.  2  x.  B )
)
31, 2mpd 15 . 2  |-  ( -.  2  x.  B  ->  -.  2  x.  B
)
4 df-or 360 . 2  |-  ( ( 2  x.  B  \/  -.  2  x.  B
)  <->  ( -.  2  x.  B  ->  -.  2  x.  B ) )
53, 4mpbir 201 1  |-  ( 2  x.  B  \/  -.  2  x.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358   class class class wbr 4212    x. cmul 8995   2c2 10049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-or 360
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