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Theorem xordc 1374
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 xornbidc 1373 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))
21imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑𝜓)))
3 excxor 1360 . . . 4 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
4 ancom 264 . . . . 5 ((¬ 𝜑𝜓) ↔ (𝜓 ∧ ¬ 𝜑))
54orbi2i 752 . . . 4 (((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
63, 5bitri 183 . . 3 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
72, 6bitr3di 194 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
87ex 114 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820  wxo 1357
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  df-xor 1358
This theorem is referenced by:  dfbi3dc  1379  pm5.24dc  1380
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