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Theorem cadbi123d 1547
Description: Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Hypotheses
Ref Expression
cadbid.1 (𝜑 → (𝜓𝜒))
cadbid.2 (𝜑 → (𝜃𝜏))
cadbid.3 (𝜑 → (𝜂𝜁))
Assertion
Ref Expression
cadbi123d (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))

Proof of Theorem cadbi123d
StepHypRef Expression
1 cadbid.1 . . . 4 (𝜑 → (𝜓𝜒))
2 cadbid.2 . . . 4 (𝜑 → (𝜃𝜏))
31, 2anbi12d 746 . . 3 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
4 cadbid.3 . . . 4 (𝜑 → (𝜂𝜁))
51, 2xorbi12d 1476 . . . 4 (𝜑 → ((𝜓𝜃) ↔ (𝜒𝜏)))
64, 5anbi12d 746 . . 3 (𝜑 → ((𝜂 ∧ (𝜓𝜃)) ↔ (𝜁 ∧ (𝜒𝜏))))
73, 6orbi12d 745 . 2 (𝜑 → (((𝜓𝜃) ∨ (𝜂 ∧ (𝜓𝜃))) ↔ ((𝜒𝜏) ∨ (𝜁 ∧ (𝜒𝜏)))))
8 df-cad 1544 . 2 (cadd(𝜓, 𝜃, 𝜂) ↔ ((𝜓𝜃) ∨ (𝜂 ∧ (𝜓𝜃))))
9 df-cad 1544 . 2 (cadd(𝜒, 𝜏, 𝜁) ↔ ((𝜒𝜏) ∨ (𝜁 ∧ (𝜒𝜏))))
107, 8, 93bitr4g 303 1 (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  wxo 1462  caddwcad 1543
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-xor 1463  df-cad 1544
This theorem is referenced by:  cadbi123i  1548  sadfval  15155  sadcp1  15158
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