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Theorem pm5.6dc 834
Description: Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 835). (Contributed by Jim Kingdon, 2-Apr-2018.)
Assertion
Ref Expression
pm5.6dc (DECID ψ → (((φ ¬ ψ) → χ) ↔ (φ → (ψ χ))))

Proof of Theorem pm5.6dc
StepHypRef Expression
1 dfordc 790 . . 3 (DECID ψ → ((ψ χ) ↔ (¬ ψχ)))
21imbi2d 219 . 2 (DECID ψ → ((φ → (ψ χ)) ↔ (φ → (¬ ψχ))))
3 impexp 250 . 2 (((φ ¬ ψ) → χ) ↔ (φ → (¬ ψχ)))
42, 3syl6rbbr 188 1 (DECID ψ → (((φ ¬ ψ) → χ) ↔ (φ → (ψ χ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  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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