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Theorem testbitestn 861
Description: A proposition is testable iff its negation is testable. See also dcn 784 (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 663 . . . 4 (¬ ¬ ¬ 𝜑 ↔ ¬ 𝜑)
21orbi2i 714 . . 3 ((¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑) ↔ (¬ ¬ 𝜑 ∨ ¬ 𝜑))
3 orcom 682 . . 3 ((¬ ¬ 𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
42, 3bitri 182 . 2 ((¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
5 df-dc 781 . 2 (DECID ¬ ¬ 𝜑 ↔ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑))
6 df-dc 781 . 2 (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
74, 5, 63bitr4ri 211 1 (DECID ¬ 𝜑DECID ¬ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wo 664  DECID wdc 780
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781
This theorem is referenced by: (None)
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