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

Theorem xordidc 1290
 Description: Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.)
Assertion
Ref Expression
xordidc (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))))

Proof of Theorem xordidc
StepHypRef Expression
1 dcbi 844 . . . . 5 (DECID 𝜓 → (DECID 𝜒DECID (𝜓𝜒)))
21imp 115 . . . 4 ((DECID 𝜓DECID 𝜒) → DECID (𝜓𝜒))
3 annimdc 845 . . . . . 6 (DECID 𝜑 → (DECID (𝜓𝜒) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))))
43imp 115 . . . . 5 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒))))
5 pm5.32 426 . . . . . 6 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
65notbii 594 . . . . 5 (¬ (𝜑 → (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
74, 6syl6bb 185 . . . 4 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
82, 7sylan2 270 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
9 xornbidc 1282 . . . . . 6 (DECID 𝜓 → (DECID 𝜒 → ((𝜓𝜒) ↔ ¬ (𝜓𝜒))))
109imp 115 . . . . 5 ((DECID 𝜓DECID 𝜒) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1110adantl 262 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1211anbi2d 437 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒))))
13 dcan 842 . . . . . 6 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
1413imp 115 . . . . 5 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1514adantrr 448 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜓))
16 dcan 842 . . . . . 6 (DECID 𝜑 → (DECID 𝜒DECID (𝜑𝜒)))
1716imp 115 . . . . 5 ((DECID 𝜑DECID 𝜒) → DECID (𝜑𝜒))
1817adantrl 447 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜒))
19 xornbidc 1282 . . . 4 (DECID (𝜑𝜓) → (DECID (𝜑𝜒) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))))
2015, 18, 19sylc 56 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
218, 12, 203bitr4d 209 . 2 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
2221exp32 347 1 (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742   ⊻ wxo 1266 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  df-xor 1267 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator