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Theorem cadnot 1617
Description: The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
Assertion
Ref Expression
cadnot (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))

Proof of Theorem cadnot
StepHypRef Expression
1 ianor 979 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2 ianor 979 . . 3 (¬ (𝜑𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜒))
3 ianor 979 . . 3 (¬ (𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
41, 2, 33anbi123i 1154 . 2 ((¬ (𝜑𝜓) ∧ ¬ (𝜑𝜒) ∧ ¬ (𝜓𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜒) ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
5 3ioran 1105 . . 3 (¬ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)) ↔ (¬ (𝜑𝜓) ∧ ¬ (𝜑𝜒) ∧ ¬ (𝜓𝜒)))
6 cador 1610 . . 3 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))
75, 6xchnxbir 333 . 2 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ (𝜑𝜒) ∧ ¬ (𝜓𝜒)))
8 cadan 1611 . 2 (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜒) ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
94, 7, 83bitr4i 303 1 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wo 844  w3o 1085  w3a 1086  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 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-cad 1609
This theorem is referenced by:  wl-df3maxtru1  35671
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