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Theorem pm5.55dc 818
Description: A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
Assertion
Ref Expression
pm5.55dc (DECID φ → (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ)))

Proof of Theorem pm5.55dc
StepHypRef Expression
1 df-dc 742 . 2 (DECID φ ↔ (φ ¬ φ))
2 biort 737 . . . 4 (φ → (φ ↔ (φ ψ)))
32bicomd 129 . . 3 (φ → ((φ ψ) ↔ φ))
4 biorf 662 . . . 4 φ → (ψ ↔ (φ ψ)))
54bicomd 129 . . 3 φ → ((φ ψ) ↔ ψ))
63, 5orim12i 675 . 2 ((φ ¬ φ) → (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ)))
71, 6sylbi 114 1 (DECID φ → (((φ ψ) ↔ φ) ((φ ψ) ↔ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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-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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