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Theorem cad1 1618
Description: If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 19-Jun-2020.)
Assertion
Ref Expression
cad1 (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑𝜓)))

Proof of Theorem cad1
StepHypRef Expression
1 olc 865 . . . 4 (𝜒 → (𝜑𝜒))
2 olc 865 . . . 4 (𝜒 → (𝜓𝜒))
31, 2jca 515 . . 3 (𝜒 → ((𝜑𝜒) ∧ (𝜓𝜒)))
43biantrud 535 . 2 (𝜒 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ ((𝜑𝜒) ∧ (𝜓𝜒)))))
5 cadan 1611 . . 3 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜓𝜒)))
6 3anass 1092 . . 3 (((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ ((𝜑𝜒) ∧ (𝜓𝜒))))
75, 6bitri 278 . 2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ ((𝜑𝜒) ∧ (𝜓𝜒))))
84, 7syl6rbbr 293 1 (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084  caddwcad 1608
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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-xor 1503  df-cad 1609
This theorem is referenced by:  cadifp  1620  sadadd2lem2  15786  sadcaddlem  15793
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