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Theorem xor2dc 1369
Description: Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
xor2dc (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))))

Proof of Theorem xor2dc
StepHypRef Expression
1 xor3dc 1366 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
21imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3 pm5.17dc 890 . . . 4 (DECID 𝜓 → (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
43adantl 275 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
52, 4bitr4d 190 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
65ex 114 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820
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 604  ax-in2 605  ax-io 699
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821
This theorem is referenced by:  xornbidc  1370
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