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Theorem hadcoma 1600
Description: Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
Assertion
Ref Expression
hadcoma (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))

Proof of Theorem hadcoma
StepHypRef Expression
1 bicom 225 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
21bibi1i 342 . 2 (((𝜑𝜓) ↔ 𝜒) ↔ ((𝜓𝜑) ↔ 𝜒))
3 hadbi 1599 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ↔ 𝜒))
4 hadbi 1599 . 2 (hadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓𝜑) ↔ 𝜒))
52, 3, 43bitr4i 306 1 (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 209  haddwhad 1594
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-xor 1503  df-had 1595
This theorem is referenced by:  hadrot  1603  sadcom  15805
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