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Theorem xordc 1387
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 1386 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))
21imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑𝜓)))
3 excxor 1373 . . . 4 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
4 ancom 264 . . . . 5 ((¬ 𝜑𝜓) ↔ (𝜓 ∧ ¬ 𝜑))
54orbi2i 757 . . . 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 703  DECID wdc 829  wxo 1370
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 609  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-xor 1371
This theorem is referenced by:  dfbi3dc  1392  pm5.24dc  1393
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