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 Description: Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
Assertion
Ref Expression

StepHypRef Expression
1 cadbid.1 . . . 4 (𝜑 → (𝜓𝜒))
2 cadbid.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2anbi12d 625 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 cadbid.3 . . . 4 (𝜑 → (𝜂𝜁))
51, 2xorbi12d 1648 . . . 4 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
64, 5anbi12d 625 . . 3 (𝜑 → ((𝜂 ∧ (𝜓𝜃)) ↔ (𝜁 ∧ (𝜒𝜏))))
73, 6orbi12d 943 . 2 (𝜑 → (((𝜓𝜃) ∨ (𝜂 ∧ (𝜓𝜃))) ↔ ((𝜒𝜏) ∨ (𝜁 ∧ (𝜒𝜏)))))
8 df-cad 1717 . 2 (cadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ∨ (𝜂 ∧ (𝜓𝜃))))
9 df-cad 1717 . 2 (cadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒𝜏) ∨ (𝜁 ∧ (𝜒𝜏))))
107, 8, 93bitr4g 306 1 (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∨ wo 874   ⊻ wxo 1634  caddwcad 1716 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 199  df-an 386  df-or 875  df-xor 1635  df-cad 1717 This theorem is referenced by:  cadbi123i  1721  sadfval  15506  sadcp1  15509
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