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Theorem testbitestn 822
Description: A proposition is testable iff its negation is testable. See also dcn 745 (which could be read as "Decidability implies testability"). (Contributed by David A. Wheeler, 6-Dec-2018.)
Assertion
Ref Expression
testbitestn (DECID ¬ φDECID ¬ ¬ φ)

Proof of Theorem testbitestn
StepHypRef Expression
1 notnotnot 627 . . . 4 (¬ ¬ ¬ φ ↔ ¬ φ)
21orbi2i 678 . . 3 ((¬ ¬ φ ¬ ¬ ¬ φ) ↔ (¬ ¬ φ ¬ φ))
3 orcom 646 . . 3 ((¬ ¬ φ ¬ φ) ↔ (¬ φ ¬ ¬ φ))
42, 3bitri 173 . 2 ((¬ ¬ φ ¬ ¬ ¬ φ) ↔ (¬ φ ¬ ¬ φ))
5 df-dc 742 . 2 (DECID ¬ ¬ φ ↔ (¬ ¬ φ ¬ ¬ ¬ φ))
6 df-dc 742 . 2 (DECID ¬ φ ↔ (¬ φ ¬ ¬ φ))
74, 5, 63bitr4ri 202 1 (DECID ¬ φDECID ¬ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   wo 628  DECID wdc 741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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