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Theorem cadcomb 1549
Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.)
Assertion
Ref Expression
cadcomb (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓))

Proof of Theorem cadcomb
StepHypRef Expression
1 cadan 1545 . . 3 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜓𝜒)))
2 3ancoma 1043 . . 3 (((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜓𝜒)) ↔ ((𝜑𝜒) ∧ (𝜑𝜓) ∧ (𝜓𝜒)))
3 orcom 402 . . . 4 ((𝜓𝜒) ↔ (𝜒𝜓))
433anbi3i 1253 . . 3 (((𝜑𝜒) ∧ (𝜑𝜓) ∧ (𝜓𝜒)) ↔ ((𝜑𝜒) ∧ (𝜑𝜓) ∧ (𝜒𝜓)))
51, 2, 43bitri 286 . 2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜒) ∧ (𝜑𝜓) ∧ (𝜒𝜓)))
6 cadan 1545 . 2 (cadd(𝜑, 𝜒, 𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜓) ∧ (𝜒𝜓)))
75, 6bitr4i 267 1 (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  w3a 1036  caddwcad 1542
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-3or 1037  df-3an 1038  df-xor 1462  df-cad 1543
This theorem is referenced by:  cadrot  1550
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