Theorem wl-hadrot 34770

Description: Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)

Assertion

Ref	Expression
wl-hadrot	⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))

Proof of Theorem wl-hadrot

Step	Hyp	Ref	Expression
1		bicom 224	. 2 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑))
2		wl-dfhad2 34764	. 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
3		wl-dfhad3 34765	. 2 ⊢ (hadd(𝜓, 𝜒, 𝜑) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑))
4	1, 2, 3	3bitr4i 305	1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))

Syntax hints:   ↔ wb 208  haddwhad 1593

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 209  df-an 399  df-or 844  df-ifp 1058  df-xor 1502  df-tru 1540  df-had 1594

This theorem is referenced by:  wl-hadcomb  34772