Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cadnot Structured version   Visualization version   GIF version

 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 509 . . 3 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))
2 ianor 509 . . 3 (¬ (𝜑𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜒))
3 ianor 509 . . 3 (¬ (𝜓𝜒) ↔ (¬ 𝜓 ∨ ¬ 𝜒))
41, 2, 33anbi123i 1250 . 2 ((¬ (𝜑𝜓) ∧ ¬ (𝜑𝜒) ∧ ¬ (𝜓𝜒)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜒) ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
5 3ioran 1055 . . 3 (¬ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)) ↔ (¬ (𝜑𝜓) ∧ ¬ (𝜑𝜒) ∧ ¬ (𝜓𝜒)))
6 cador 1546 . . 3 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜑𝜒) ∨ (𝜓𝜒)))
75, 6xchnxbir 323 . 2 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ (¬ (𝜑𝜓) ∧ ¬ (𝜑𝜒) ∧ ¬ (𝜓𝜒)))
8 cadan 1547 . 2 (cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∧ (¬ 𝜑 ∨ ¬ 𝜒) ∧ (¬ 𝜓 ∨ ¬ 𝜒)))
94, 7, 83bitr4i 292 1 (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037  caddwcad 1544 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 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1464  df-cad 1545 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator