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Theorem pm5.17dc 810
 Description: Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
Assertion
Ref Expression
pm5.17dc (DECID 𝜓 → (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))

Proof of Theorem pm5.17dc
StepHypRef Expression
1 bicom 128 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝜑))
2 dfbi2 368 . . 3 ((¬ 𝜓𝜑) ↔ ((¬ 𝜓𝜑) ∧ (𝜑 → ¬ 𝜓)))
3 orcom 647 . . . . 5 ((𝜑𝜓) ↔ (𝜓𝜑))
4 dfordc 791 . . . . 5 (DECID 𝜓 → ((𝜓𝜑) ↔ (¬ 𝜓𝜑)))
53, 4syl5rbb 182 . . . 4 (DECID 𝜓 → ((¬ 𝜓𝜑) ↔ (𝜑𝜓)))
6 imnan 624 . . . . 5 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
76a1i 9 . . . 4 (DECID 𝜓 → ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
85, 7anbi12d 442 . . 3 (DECID 𝜓 → (((¬ 𝜓𝜑) ∧ (𝜑 → ¬ 𝜓)) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
92, 8syl5bb 181 . 2 (DECID 𝜓 → ((¬ 𝜓𝜑) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
101, 9syl5rbb 182 1 (DECID 𝜓 → (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  xor2dc  1281
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