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 Description: Write the adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression

StepHypRef Expression
1 df-cad 1381 . 2 (cadd(φ, ψ, χ) ↔ ((φ ψ) (χ (φψ))))
2 xor2 1310 . . . . . . 7 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
32rbaib 873 . . . . . 6 (¬ (φ ψ) → ((φψ) ↔ (φ ψ)))
43anbi1d 685 . . . . 5 (¬ (φ ψ) → (((φψ) χ) ↔ ((φ ψ) χ)))
5 ancom 437 . . . . 5 (((φψ) χ) ↔ (χ (φψ)))
6 andir 838 . . . . 5 (((φ ψ) χ) ↔ ((φ χ) (ψ χ)))
74, 5, 63bitr3g 278 . . . 4 (¬ (φ ψ) → ((χ (φψ)) ↔ ((φ χ) (ψ χ))))
87pm5.74i 236 . . 3 ((¬ (φ ψ) → (χ (φψ))) ↔ (¬ (φ ψ) → ((φ χ) (ψ χ))))
9 df-or 359 . . 3 (((φ ψ) (χ (φψ))) ↔ (¬ (φ ψ) → (χ (φψ))))
10 3orass 937 . . . 4 (((φ ψ) (φ χ) (ψ χ)) ↔ ((φ ψ) ((φ χ) (ψ χ))))
11 df-or 359 . . . 4 (((φ ψ) ((φ χ) (ψ χ))) ↔ (¬ (φ ψ) → ((φ χ) (ψ χ))))
1210, 11bitri 240 . . 3 (((φ ψ) (φ χ) (ψ χ)) ↔ (¬ (φ ψ) → ((φ χ) (ψ χ))))
138, 9, 123bitr4i 268 . 2 (((φ ψ) (χ (φψ))) ↔ ((φ ψ) (φ χ) (ψ χ)))
141, 13bitri 240 1 (cadd(φ, ψ, χ) ↔ ((φ ψ) (φ χ) (ψ χ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   ⊻ wxo 1304  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-xor 1305  df-cad 1381 This theorem is referenced by:  cadan  1392  cadnot  1394  cadcomb  1396
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