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Theorem xordidc 1335
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 882 . . . . 5 (DECID 𝜓 → (DECID 𝜒DECID (𝜓𝜒)))
21imp 122 . . . 4 ((DECID 𝜓DECID 𝜒) → DECID (𝜓𝜒))
3 annimdc 883 . . . . . 6 (DECID 𝜑 → (DECID (𝜓𝜒) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))))
43imp 122 . . . . 5 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒))))
5 pm5.32 441 . . . . . 6 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
65notbii 629 . . . . 5 (¬ (𝜑 → (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
74, 6syl6bb 194 . . . 4 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
82, 7sylan2 280 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
9 xornbidc 1327 . . . . . 6 (DECID 𝜓 → (DECID 𝜒 → ((𝜓𝜒) ↔ ¬ (𝜓𝜒))))
109imp 122 . . . . 5 ((DECID 𝜓DECID 𝜒) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1110adantl 271 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1211anbi2d 452 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒))))
13 dcan 880 . . . . . 6 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
1413imp 122 . . . . 5 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1514adantrr 463 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜓))
16 dcan 880 . . . . . 6 (DECID 𝜑 → (DECID 𝜒DECID (𝜑𝜒)))
1716imp 122 . . . . 5 ((DECID 𝜑DECID 𝜒) → DECID (𝜑𝜒))
1817adantrl 462 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜒))
19 xornbidc 1327 . . . 4 (DECID (𝜑𝜓) → (DECID (𝜑𝜒) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))))
2015, 18, 19sylc 61 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
218, 12, 203bitr4d 218 . 2 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
2221exp32 357 1 (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  DECID wdc 780  wxo 1311
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 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-xor 1312
This theorem is referenced by: (None)
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