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Theorem pm5.15dc 1368
Description: A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
pm5.15dc (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))

Proof of Theorem pm5.15dc
StepHypRef Expression
1 xor3dc 1366 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
21imp 123 . . . 4 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
32biimpd 143 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) → (𝜑 ↔ ¬ 𝜓)))
4 dcbi 921 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
54imp 123 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
6 dfordc 878 . . . 4 (DECID (𝜑𝜓) → (((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) ↔ (¬ (𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))))
75, 6syl 14 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) ↔ (¬ (𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))))
83, 7mpbird 166 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓)))
98ex 114 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821
This theorem is referenced by: (None)
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