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Theorem pm5.55dc 830
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 → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))

Proof of Theorem pm5.55dc
StepHypRef Expression
1 df-dc 754 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 biort 749 . . . 4 (𝜑 → (𝜑 ↔ (𝜑𝜓)))
32bicomd 133 . . 3 (𝜑 → ((𝜑𝜓) ↔ 𝜑))
4 biorf 673 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
54bicomd 133 . . 3 𝜑 → ((𝜑𝜓) ↔ 𝜓))
63, 5orim12i 686 . 2 ((𝜑 ∨ ¬ 𝜑) → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))
71, 6sylbi 118 1 (DECID 𝜑 → (((𝜑𝜓) ↔ 𝜑) ∨ ((𝜑𝜓) ↔ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
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