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Theorem cadcoma 1700
Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
cadcoma (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒))

Proof of Theorem cadcoma
StepHypRef Expression
1 ancom 465 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
2 xorcom 1616 . . . 4 ((𝜑𝜓) ↔ (𝜓𝜑))
32anbi2i 732 . . 3 ((𝜒 ∧ (𝜑𝜓)) ↔ (𝜒 ∧ (𝜓𝜑)))
41, 3orbi12i 544 . 2 (((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))) ↔ ((𝜓𝜑) ∨ (𝜒 ∧ (𝜓𝜑))))
5 df-cad 1695 . 2 (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (𝜒 ∧ (𝜑𝜓))))
6 df-cad 1695 . 2 (cadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓𝜑) ∨ (𝜒 ∧ (𝜓𝜑))))
74, 5, 63bitr4i 292 1 (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383  wxo 1613  caddwcad 1694
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 384  df-an 385  df-xor 1614  df-cad 1695
This theorem is referenced by:  cadrot  1702  sadcom  15407
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