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Theorem xordidc 1394
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 931 . . . . 5 (DECID 𝜓 → (DECID 𝜒DECID (𝜓𝜒)))
21imp 123 . . . 4 ((DECID 𝜓DECID 𝜒) → DECID (𝜓𝜒))
3 annimdc 932 . . . . . 6 (DECID 𝜑 → (DECID (𝜓𝜒) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒)))))
43imp 123 . . . . 5 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ (𝜑 → (𝜓𝜒))))
5 pm5.32 450 . . . . . 6 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) ↔ (𝜑𝜒)))
65notbii 663 . . . . 5 (¬ (𝜑 → (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))
74, 6bitrdi 195 . . . 4 ((DECID 𝜑DECID (𝜓𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
82, 7sylan2 284 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ ¬ (𝜓𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
9 xornbidc 1386 . . . . . 6 (DECID 𝜓 → (DECID 𝜒 → ((𝜓𝜒) ↔ ¬ (𝜓𝜒))))
109imp 123 . . . . 5 ((DECID 𝜓DECID 𝜒) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1110adantl 275 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜓𝜒) ↔ ¬ (𝜓𝜒)))
1211anbi2d 461 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ (𝜑 ∧ ¬ (𝜓𝜒))))
13 dcan2 929 . . . . . 6 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
1413imp 123 . . . . 5 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
1514adantrr 476 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜓))
16 dcan2 929 . . . . . 6 (DECID 𝜑 → (DECID 𝜒DECID (𝜑𝜒)))
1716imp 123 . . . . 5 ((DECID 𝜑DECID 𝜒) → DECID (𝜑𝜒))
1817adantrl 475 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → DECID (𝜑𝜒))
19 xornbidc 1386 . . . 4 (DECID (𝜑𝜓) → (DECID (𝜑𝜒) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒)))))
2015, 18, 19sylc 62 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → (((𝜑𝜓) ⊻ (𝜑𝜒)) ↔ ¬ ((𝜑𝜓) ↔ (𝜑𝜒))))
218, 12, 203bitr4d 219 . 2 ((DECID 𝜑 ∧ (DECID 𝜓DECID 𝜒)) → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))
2221exp32 363 1 (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ⊻ (𝜑𝜒))))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  DECID wdc 829  wxo 1370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-xor 1371
This theorem is referenced by: (None)
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