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Theorem condandc 867
 Description: Proof by contradiction. This only holds for decidable propositions, as it is part of the family of theorems which assume , derive a contradiction, and therefore conclude . By contrast, assuming , deriving a contradiction, and therefore concluding , as in pm2.65 649, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
Hypotheses
Ref Expression
condandc.1
condandc.2
Assertion
Ref Expression
condandc DECID

Proof of Theorem condandc
StepHypRef Expression
1 condandc.1 . . 3
2 condandc.2 . . 3
31, 2pm2.65da 651 . 2
4 notnotrdc 829 . 2 DECID
53, 4syl5 32 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103  DECID wdc 820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-dc 821 This theorem is referenced by: (None)
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