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Theorem dfbi3dc 1375
 Description: An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
dfbi3dc (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))

Proof of Theorem dfbi3dc
StepHypRef Expression
1 dcn 827 . . . 4 (DECID 𝜓DECID ¬ 𝜓)
2 xordc 1370 . . . . 5 (DECID 𝜑 → (DECID ¬ 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))))
32imp 123 . . . 4 ((DECID 𝜑DECID ¬ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
41, 3sylan2 284 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
5 pm5.18dc 868 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
65imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
7 notnotbdc 857 . . . . . 6 (DECID 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
87anbi2d 459 . . . . 5 (DECID 𝜓 → ((𝜑𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓)))
9 ancom 264 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
109a1i 9 . . . . 5 (DECID 𝜓 → ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑)))
118, 10orbi12d 782 . . . 4 (DECID 𝜓 → (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
1211adantl 275 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
134, 6, 123bitr4d 219 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
1413ex 114 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819 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 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-xor 1354 This theorem is referenced by:  pm5.24dc  1376
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