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Theorem pm5.62dc 851
Description: Theorem *5.62 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.62dc (DECID ψ → (((φ ψ) ¬ ψ) ↔ (φ ¬ ψ)))

Proof of Theorem pm5.62dc
StepHypRef Expression
1 df-dc 742 . 2 (DECID ψ ↔ (ψ ¬ ψ))
2 ordir 729 . . . 4 (((φ ψ) ¬ ψ) ↔ ((φ ¬ ψ) (ψ ¬ ψ)))
32simplbi 259 . . 3 (((φ ψ) ¬ ψ) → (φ ¬ ψ))
42simplbi2 367 . . . 4 ((φ ¬ ψ) → ((ψ ¬ ψ) → ((φ ψ) ¬ ψ)))
54com12 27 . . 3 ((ψ ¬ ψ) → ((φ ¬ ψ) → ((φ ψ) ¬ ψ)))
63, 5impbid2 131 . 2 ((ψ ¬ ψ) → (((φ ψ) ¬ ψ) ↔ (φ ¬ ψ)))
71, 6sylbi 114 1 (DECID ψ → (((φ ψ) ¬ 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-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by: (None)
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