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Theorem wl-df2-3mintru2 34895
Description: The adder carry in disjunctive normal form. An alternative highly symmetric definition emphasizing the independance of order of the inputs 𝜑, 𝜓 and 𝜒. Copy of cador 1610. (Contributed by Mario Carneiro, 4-Sep-2016.) df-cad redefined. (Revised by Wolf Lammen, 12-Jun-2024.)
Assertion
Ref Expression
wl-df2-3mintru2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))

Proof of Theorem wl-df2-3mintru2
StepHypRef Expression
1 andi 1005 . . 3 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))
21orbi1i 911 . 2 (((𝜑 ∧ (𝜓𝜒)) ∨ (𝜓𝜒)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (𝜓𝜒)))
3 wl-df-3mintru2 34894 . . 3 (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓𝜒), (𝜓𝜒)))
4 animorl 975 . . . 4 ((𝜓𝜒) → (𝜓𝜒))
5 wl-ifpimpr 34876 . . . 4 (((𝜓𝜒) → (𝜓𝜒)) → (if-(𝜑, (𝜓𝜒), (𝜓𝜒)) ↔ ((𝜑 ∧ (𝜓𝜒)) ∨ (𝜓𝜒))))
64, 5ax-mp 5 . . 3 (if-(𝜑, (𝜓𝜒), (𝜓𝜒)) ↔ ((𝜑 ∧ (𝜓𝜒)) ∨ (𝜓𝜒)))
73, 6bitri 278 . 2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ (𝜓𝜒)) ∨ (𝜓𝜒)))
8 df-3or 1085 . 2 (((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)) ↔ (((𝜑𝜓) ∨ (𝜑𝜒)) ∨ (𝜓𝜒)))
92, 7, 83bitr4i 306 1 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  if-wif 1058  w3o 1083  caddwcad 1608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-xor 1503  df-cad 1609
This theorem is referenced by:  wl-df4-3mintru2  34897
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