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Theorem dcimptest 744
 Description: Decidability implies testability. (Contributed by David A. Wheeler, 14-Aug-2018.)
Assertion
Ref Expression
dcimptest (DECID φTEST φ)

Proof of Theorem dcimptest
StepHypRef Expression
1 notnot1 547 . . . 4 (φ → ¬ ¬ φ)
21orim1i 664 . . 3 ((φ ¬ φ) → (¬ ¬ φ ¬ φ))
32orcomd 635 . 2 ((φ ¬ φ) → (¬ φ ¬ ¬ φ))
4 df-dc 734 . 2 (DECID φ ↔ (φ ¬ φ))
5 df-test 731 . 2 (TEST φ ↔ (¬ φ ¬ ¬ φ))
63, 4, 53imtr4i 190 1 (DECID φTEST φ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 616  TEST wtest 730  DECID wdc 733 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 532  ax-in2 533  ax-io 617 This theorem depends on definitions:  df-bi 110  df-test 731  df-dc 734 This theorem is referenced by:  stabtestimpdc  745
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