Description: Commutative law for triple xor. Copy of hadcoma . (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 17-Dec-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | wl-3xorcoma | ⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ hadd ( 𝜓 , 𝜑 , 𝜒 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom | ⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) ) | |
2 | 1 | bibi1i | ⊢ ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( ( 𝜓 ↔ 𝜑 ) ↔ 𝜒 ) ) |
3 | wl-3xorbi2 | ⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ) | |
4 | wl-3xorbi2 | ⊢ ( hadd ( 𝜓 , 𝜑 , 𝜒 ) ↔ ( ( 𝜓 ↔ 𝜑 ) ↔ 𝜒 ) ) | |
5 | 2 3 4 | 3bitr4i | ⊢ ( hadd ( 𝜑 , 𝜓 , 𝜒 ) ↔ hadd ( 𝜓 , 𝜑 , 𝜒 ) ) |