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Theorem xordc1 1354
Description: Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
Assertion
Ref Expression
xordc1 ((𝜑𝜓) → DECID 𝜑)

Proof of Theorem xordc1
StepHypRef Expression
1 andir 791 . . 3 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ ((𝜑 ∧ ¬ (𝜑𝜓)) ∨ (𝜓 ∧ ¬ (𝜑𝜓))))
2 simpl 108 . . . 4 ((𝜑 ∧ ¬ (𝜑𝜓)) → 𝜑)
3 imnan 662 . . . . . 6 ((𝜓 → ¬ 𝜑) ↔ ¬ (𝜓𝜑))
4 ancom 264 . . . . . 6 ((𝜑𝜓) ↔ (𝜓𝜑))
53, 4xchbinxr 655 . . . . 5 ((𝜓 → ¬ 𝜑) ↔ ¬ (𝜑𝜓))
6 pm3.35 342 . . . . 5 ((𝜓 ∧ (𝜓 → ¬ 𝜑)) → ¬ 𝜑)
75, 6sylan2br 284 . . . 4 ((𝜓 ∧ ¬ (𝜑𝜓)) → ¬ 𝜑)
82, 7orim12i 731 . . 3 (((𝜑 ∧ ¬ (𝜑𝜓)) ∨ (𝜓 ∧ ¬ (𝜑𝜓))) → (𝜑 ∨ ¬ 𝜑))
91, 8sylbi 120 . 2 (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) → (𝜑 ∨ ¬ 𝜑))
10 df-xor 1337 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓)))
11 df-dc 803 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
129, 10, 113imtr4i 200 1 ((𝜑𝜓) → DECID 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 680  DECID wdc 802  wxo 1336
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 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-dc 803  df-xor 1337
This theorem is referenced by: (None)
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