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Theorem bj-stdc 14752
Description: Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)
Assertion
Ref Expression
bj-stdc (STAB DECID 𝜑DECID 𝜑)

Proof of Theorem bj-stdc
StepHypRef Expression
1 nndc 852 . 2 ¬ ¬ DECID 𝜑
2 bj-nnbist 14736 . 2 (¬ ¬ DECID 𝜑 → (STAB DECID 𝜑DECID 𝜑))
31, 2ax-mp 5 1 (STAB DECID 𝜑DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  STAB wstab 831  DECID wdc 835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836
This theorem is referenced by: (None)
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