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Theorem xor3dc 1278
 Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
Assertion
Ref Expression
xor3dc (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))

Proof of Theorem xor3dc
StepHypRef Expression
1 dcn 746 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
2 dcbi 844 . . . . . 6 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ↔ ¬ 𝜓)))
31, 2syl5 28 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑 ↔ ¬ 𝜓)))
43imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ↔ ¬ 𝜓))
5 pm5.18dc 777 . . . . . . 7 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
65imp 115 . . . . . 6 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
76a1d 22 . . . . 5 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
87con2biddc 774 . . . 4 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
94, 8mpd 13 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
109bicomd 129 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
1110ex 108 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  pm5.15dc  1280  xor2dc  1281  nbbndc  1285
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