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Theorem pm2.65 631
Description: Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here  ph, derive a contradiction, and therefore conclude  -. 
ph, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume  -.  ph, derive a contradiction, and conclude  ph, such as condandc 849, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)
Assertion
Ref Expression
pm2.65  |-  ( (
ph  ->  ps )  -> 
( ( ph  ->  -. 
ps )  ->  -.  ph ) )

Proof of Theorem pm2.65
StepHypRef Expression
1 pm2.27 40 . . . 4  |-  ( ph  ->  ( ( ph  ->  -. 
ps )  ->  -.  ps ) )
21con2d 596 . . 3  |-  ( ph  ->  ( ps  ->  -.  ( ph  ->  -.  ps )
) )
32a2i 11 . 2  |-  ( (
ph  ->  ps )  -> 
( ph  ->  -.  ( ph  ->  -.  ps )
) )
43con2d 596 1  |-  ( (
ph  ->  ps )  -> 
( ( ph  ->  -. 
ps )  ->  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 586  ax-in2 587
This theorem is referenced by:  pm4.82  917
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