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Theorem hadcoma 1578
Description: Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadcoma (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))

Proof of Theorem hadcoma
StepHypRef Expression
1 xorcom 1507 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
2 biid 251 . . 3 (𝜒𝜒)
31, 2xorbi12i 1517 . 2 (((𝜑𝜓) ⊻ 𝜒) ↔ ((𝜓𝜑) ⊻ 𝜒))
4 df-had 1573 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ⊻ 𝜒))
5 df-had 1573 . 2 (hadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓𝜑) ⊻ 𝜒))
63, 4, 53bitr4i 292 1 (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wxo 1504  haddwhad 1572
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-xor 1505  df-had 1573
This theorem is referenced by:  hadrot  1580  sadcom  15232
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