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Theorem xordc 1328
Description: Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
Assertion
Ref Expression
xordc (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))

Proof of Theorem xordc
StepHypRef Expression
1 excxor 1314 . . . 4 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
2 ancom 262 . . . . 5 ((¬ 𝜑𝜓) ↔ (𝜓 ∧ ¬ 𝜑))
32orbi2i 714 . . . 4 (((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
41, 3bitri 182 . . 3 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
5 xornbidc 1327 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))
65imp 122 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑𝜓)))
74, 6syl5rbbr 193 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
87ex 113 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780  wxo 1311
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-xor 1312
This theorem is referenced by:  dfbi3dc  1333  pm5.24dc  1334
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