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 Description: The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression

StepHypRef Expression
1 3ioran 950 . . 3 (¬ ((φ ψ) (φ χ) (ψ χ)) ↔ (¬ (φ ψ) ¬ (φ χ) ¬ (ψ χ)))
2 ianor 474 . . . 4 (¬ (φ ψ) ↔ (¬ φ ¬ ψ))
3 ianor 474 . . . 4 (¬ (φ χ) ↔ (¬ φ ¬ χ))
4 ianor 474 . . . 4 (¬ (ψ χ) ↔ (¬ ψ ¬ χ))
52, 3, 43anbi123i 1140 . . 3 ((¬ (φ ψ) ¬ (φ χ) ¬ (ψ χ)) ↔ ((¬ φ ¬ ψ) φ ¬ χ) ψ ¬ χ)))
61, 5bitri 240 . 2 (¬ ((φ ψ) (φ χ) (ψ χ)) ↔ ((¬ φ ¬ ψ) φ ¬ χ) ψ ¬ χ)))
7 cador 1391 . . 3 (cadd(φ, ψ, χ) ↔ ((φ ψ) (φ χ) (ψ χ)))
87notbii 287 . 2 (¬ cadd(φ, ψ, χ) ↔ ¬ ((φ ψ) (φ χ) (ψ χ)))
9 cadan 1392 . 2 (cadd(¬ φ, ¬ ψ, ¬ χ) ↔ ((¬ φ ¬ ψ) φ ¬ χ) ψ ¬ χ)))
106, 8, 93bitr4i 268 1 (¬ cadd(φ, ψ, χ) ↔ cadd(¬ φ, ¬ ψ, ¬ χ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   ∧ w3a 934  caddwcad 1379 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 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1305  df-cad 1381 This theorem is referenced by: (None)
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