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Theorem pm5.15dc 1294
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 1292 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
21imp 119 . . . 4 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
32biimpd 136 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) → (𝜑 ↔ ¬ 𝜓)))
4 dcbi 853 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
54imp 119 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
6 dfordc 800 . . . 4 (DECID (𝜑𝜓) → (((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) ↔ (¬ (𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))))
75, 6syl 14 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓)) ↔ (¬ (𝜑𝜓) → (𝜑 ↔ ¬ 𝜓))))
83, 7mpbird 160 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓)))
98ex 112 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ∨ (𝜑 ↔ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 637  DECID wdc 751
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-in1 552  ax-in2 553  ax-io 638
This theorem depends on definitions:  df-bi 114  df-dc 752
This theorem is referenced by: (None)
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