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Theorem pm3.13dc 865
 Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 669, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.13dc (DECID φ → (DECID ψ → (¬ (φ ψ) → (¬ φ ¬ ψ))))

Proof of Theorem pm3.13dc
StepHypRef Expression
1 dcn 745 . . 3 (DECID φDECID ¬ φ)
2 dcn 745 . . 3 (DECID ψDECID ¬ ψ)
3 dcor 842 . . 3 (DECID ¬ φ → (DECID ¬ ψDECIDφ ¬ ψ)))
41, 2, 3syl2im 34 . 2 (DECID φ → (DECID ψDECIDφ ¬ ψ)))
5 pm3.11dc 863 . 2 (DECID φ → (DECID ψ → (¬ (¬ φ ¬ ψ) → (φ ψ))))
6 con1dc 752 . 2 (DECIDφ ¬ ψ) → ((¬ (¬ φ ¬ ψ) → (φ ψ)) → (¬ (φ ψ) → (¬ φ ¬ ψ))))
74, 5, 6syl6c 60 1 (DECID φ → (DECID ψ → (¬ (φ ψ) → (¬ φ ¬ ψ))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ 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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