ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xor3dc GIF version

Theorem xor3dc 1319
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 780 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
2 dcbi 878 . . . . . 6 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ↔ ¬ 𝜓)))
31, 2syl5 32 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑 ↔ ¬ 𝜓)))
43imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ↔ ¬ 𝜓))
5 pm5.18dc 811 . . . . . . 7 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
65imp 122 . . . . . 6 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
76a1d 22 . . . . 5 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
87con2biddc 808 . . . 4 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ↔ ¬ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
94, 8mpd 13 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
109bicomd 139 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
1110ex 113 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  DECID wdc 776
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 777
This theorem is referenced by:  pm5.15dc  1321  xor2dc  1322  nbbndc  1326
  Copyright terms: Public domain W3C validator