ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.17dc GIF version

Theorem pm5.17dc 846
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 138 . 2 ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝜑))
2 dfbi2 380 . . 3 ((¬ 𝜓𝜑) ↔ ((¬ 𝜓𝜑) ∧ (𝜑 → ¬ 𝜓)))
3 orcom 680 . . . . 5 ((𝜑𝜓) ↔ (𝜓𝜑))
4 dfordc 827 . . . . 5 (DECID 𝜓 → ((𝜓𝜑) ↔ (¬ 𝜓𝜑)))
53, 4syl5rbb 191 . . . 4 (DECID 𝜓 → ((¬ 𝜓𝜑) ↔ (𝜑𝜓)))
6 imnan 657 . . . . 5 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
76a1i 9 . . . 4 (DECID 𝜓 → ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
85, 7anbi12d 457 . . 3 (DECID 𝜓 → (((¬ 𝜓𝜑) ∧ (𝜑 → ¬ 𝜓)) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
92, 8syl5bb 190 . 2 (DECID 𝜓 → ((¬ 𝜓𝜑) ↔ ((𝜑𝜓) ∧ ¬ (𝜑𝜓))))
101, 9syl5rbb 191 1 (DECID 𝜓 → (((𝜑𝜓) ∧ ¬ (𝜑𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779
This theorem is referenced by:  xor2dc  1324
  Copyright terms: Public domain W3C validator