MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hadass Structured version   Visualization version   GIF version

Theorem hadass 1603
Description: Associative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.)
Assertion
Ref Expression
hadass (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))

Proof of Theorem hadass
StepHypRef Expression
1 df-had 1600 . 2 (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ⊻ 𝜒))
2 xorass 1512 . 2 (((𝜑𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
31, 2bitri 278 1 (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wxo 1507  haddwhad 1599
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 1508  df-had 1600
This theorem is referenced by:  hadcomb  1607
  Copyright terms: Public domain W3C validator