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Theorem xordidc 1306
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 855 . . . . 5 (DECID 𝜓 → (DECID 𝜒DECID (𝜓𝜒)))
21imp 119 . . . 4 ((DECID 𝜓DECID 𝜒) → DECID (𝜓𝜒))
3 annimdc 856 . . . . . 6 (DECID 𝜑 → (DECID (𝜓𝜒) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))))
43imp 119 . . . . 5 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒))))
5 pm5.32 434 . . . . . 6 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
65notbii 604 . . . . 5 (¬ (𝜑 → (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
74, 6syl6bb 189 . . . 4 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
82, 7sylan2 274 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
9 xornbidc 1298 . . . . . 6 (DECID 𝜓 → (DECID 𝜒 → ((𝜓𝜒) ↔ ¬ (𝜓𝜒))))
109imp 119 . . . . 5 ((DECID 𝜓DECID 𝜒) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1110adantl 266 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1211anbi2d 445 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒))))
13 dcan 853 . . . . . 6 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
1413imp 119 . . . . 5 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1514adantrr 456 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜓))
16 dcan 853 . . . . . 6 (DECID 𝜑 → (DECID 𝜒DECID (𝜑𝜒)))
1716imp 119 . . . . 5 ((DECID 𝜑DECID 𝜒) → DECID (𝜑𝜒))
1817adantrl 455 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜒))
19 xornbidc 1298 . . . 4 (DECID (𝜑𝜓) → (DECID (𝜑𝜒) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))))
2015, 18, 19sylc 60 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
218, 12, 203bitr4d 213 . 2 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
2221exp32 351 1 (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  DECID wdc 753  wxo 1282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754  df-xor 1283
This theorem is referenced by: (None)
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