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Theorem dfbi3dc 1331
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 782 . . . 4 (DECID 𝜓DECID ¬ 𝜓)
2 xordc 1326 . . . . 5 (DECID 𝜑 → (DECID ¬ 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑)))))
32imp 122 . . . 4 ((DECID 𝜑DECID ¬ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
41, 3sylan2 280 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
5 pm5.18dc 813 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
65imp 122 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
7 notnotbdc 802 . . . . . 6 (DECID 𝜓 → (𝜓 ↔ ¬ ¬ 𝜓))
87anbi2d 452 . . . . 5 (DECID 𝜓 → ((𝜑𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓)))
9 ancom 262 . . . . . 6 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑))
109a1i 9 . . . . 5 (DECID 𝜓 → ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑)))
118, 10orbi12d 740 . . . 4 (DECID 𝜓 → (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
1211adantl 271 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))))
134, 6, 123bitr4d 218 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))))
1413ex 113 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779  df-xor 1310
This theorem is referenced by:  pm5.24dc  1332
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