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Theorem xornbidc 1391
Description: Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
Assertion
Ref Expression
xornbidc (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))

Proof of Theorem xornbidc
StepHypRef Expression
1 df-xor 1376 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
2 xor2dc 1390 . . . 4 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))))
32imp 124 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
41, 3bitr4id 199 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑𝜓)))
54ex 115 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  wxo 1375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-xor 1376
This theorem is referenced by:  xordc  1392  xordidc  1399
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