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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 743 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 cadbid.3 . . . 4 (𝜑 → (𝜂𝜁))
51, 2xorbi12d 1470 . . . 4 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
64, 5anbi12d 743 . . 3 (𝜑 → ((𝜂 ∧ (𝜓𝜃)) ↔ (𝜁 ∧ (𝜒𝜏))))
73, 6orbi12d 742 . 2 (𝜑 → (((𝜓𝜃) ∨ (𝜂 ∧ (𝜓𝜃))) ↔ ((𝜒𝜏) ∨ (𝜁 ∧ (𝜒𝜏)))))
8 df-cad 1537 . 2 (cadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ∨ (𝜂 ∧ (𝜓𝜃))))
9 df-cad 1537 . 2 (cadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒𝜏) ∨ (𝜁 ∧ (𝜒𝜏))))
107, 8, 93bitr4g 302 1 (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ⊻ wxo 1456  caddwcad 1536 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 196  df-or 384  df-an 385  df-xor 1457  df-cad 1537 This theorem is referenced by:  cadbi123i  1541  sadfval  15012  sadcp1  15015
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