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Theorem pm5.63dc 852
 Description: Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)
Assertion
Ref Expression
pm5.63dc (DECID φ → ((φ ψ) ↔ (φ φ ψ))))

Proof of Theorem pm5.63dc
StepHypRef Expression
1 df-dc 742 . . 3 (DECID φ ↔ (φ ¬ φ))
2 ordi 728 . . . 4 ((φ φ ψ)) ↔ ((φ ¬ φ) (φ ψ)))
32simplbi2 367 . . 3 ((φ ¬ φ) → ((φ ψ) → (φ φ ψ))))
41, 3sylbi 114 . 2 (DECID φ → ((φ ψ) → (φ φ ψ))))
52simprbi 260 . 2 ((φ φ ψ)) → (φ ψ))
64, 5impbid1 130 1 (DECID φ → ((φ ψ) ↔ (φ 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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