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Theorem testbitestn 857
Description: A proposition is testable iff its negation is testable. See also dcn 780 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)
Assertion
Ref Expression
testbitestn  |-  (DECID  -.  ph  <-> DECID  -.  -.  ph )

Proof of Theorem testbitestn
StepHypRef Expression
1 notnotnot 661 . . . 4  |-  ( -. 
-.  -.  ph  <->  -.  ph )
21orbi2i 712 . . 3  |-  ( ( -.  -.  ph  \/  -.  -.  -.  ph )  <->  ( -.  -.  ph  \/  -.  ph ) )
3 orcom 680 . . 3  |-  ( ( -.  -.  ph  \/  -.  ph )  <->  ( -.  ph  \/  -.  -.  ph ) )
42, 3bitri 182 . 2  |-  ( ( -.  -.  ph  \/  -.  -.  -.  ph )  <->  ( -.  ph  \/  -.  -.  ph ) )
5 df-dc 777 . 2  |-  (DECID  -.  -.  ph  <->  ( -.  -.  ph  \/  -.  -.  -.  ph )
)
6 df-dc 777 . 2  |-  (DECID  -.  ph  <->  ( -.  ph  \/  -.  -.  ph ) )
74, 5, 63bitr4ri 211 1  |-  (DECID  -.  ph  <-> DECID  -.  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103    \/ wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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