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Theorem pm3.12dc 864
 Description: Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
pm3.12dc (DECID φ → (DECID ψ → ((¬ φ ¬ ψ) (φ ψ))))

Proof of Theorem pm3.12dc
StepHypRef Expression
1 pm3.11dc 863 . . . 4 (DECID φ → (DECID ψ → (¬ (¬ φ ¬ ψ) → (φ ψ))))
21imp 115 . . 3 ((DECID φ DECID ψ) → (¬ (¬ φ ¬ ψ) → (φ ψ)))
3 dcn 745 . . . . . 6 (DECID φDECID ¬ φ)
4 dcn 745 . . . . . 6 (DECID ψDECID ¬ ψ)
5 dcor 842 . . . . . 6 (DECID ¬ φ → (DECID ¬ ψDECIDφ ¬ ψ)))
63, 4, 5syl2im 34 . . . . 5 (DECID φ → (DECID ψDECIDφ ¬ ψ)))
7 dfordc 790 . . . . 5 (DECIDφ ¬ ψ) → (((¬ φ ¬ ψ) (φ ψ)) ↔ (¬ (¬ φ ¬ ψ) → (φ ψ))))
86, 7syl6 29 . . . 4 (DECID φ → (DECID ψ → (((¬ φ ¬ ψ) (φ ψ)) ↔ (¬ (¬ φ ¬ ψ) → (φ ψ)))))
98imp 115 . . 3 ((DECID φ DECID ψ) → (((¬ φ ¬ ψ) (φ ψ)) ↔ (¬ (¬ φ ¬ ψ) → (φ ψ))))
102, 9mpbird 156 . 2 ((DECID φ DECID ψ) → ((¬ φ ¬ ψ) (φ ψ)))
1110ex 108 1 (DECID φ → (DECID ψ → ((¬ φ ¬ ψ) (φ ψ))))
 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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