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Theorem xor3dc 1350
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 812 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
2 dcbi 905 . . . . . 6 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ↔ ¬ 𝜓)))
31, 2syl5 32 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑 ↔ ¬ 𝜓)))
43imp 123 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ↔ ¬ 𝜓))
5 pm5.18dc 853 . . . . . . 7 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
65imp 123 . . . . . 6 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
76a1d 22 . . . . 5 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
87con2biddc 850 . . . 4 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
94, 8mpd 13 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
109bicomd 140 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
1110ex 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  DECID wdc 804
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 588  ax-in2 589  ax-io 683
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805
This theorem is referenced by:  pm5.15dc  1352  xor2dc  1353  nbbndc  1357
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