Description: Equivalence theorem for triple xor. (Contributed by Mario Carneiro, 4-Sep-2016) df-had redefined. (Revised by Wolf Lammen, 24-Apr-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | wl-3xorbid.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
wl-3xorbid.2 | ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) ) | ||
wl-3xorbid.3 | ⊢ ( 𝜑 → ( 𝜂 ↔ 𝜁 ) ) | ||
Assertion | wl-3xorbi123d | ⊢ ( 𝜑 → ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-3xorbid.1 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
2 | wl-3xorbid.2 | ⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) ) | |
3 | wl-3xorbid.3 | ⊢ ( 𝜑 → ( 𝜂 ↔ 𝜁 ) ) | |
4 | 1 2 | bibi12d | ⊢ ( 𝜑 → ( ( 𝜓 ↔ 𝜃 ) ↔ ( 𝜒 ↔ 𝜏 ) ) ) |
5 | 4 3 | bibi12d | ⊢ ( 𝜑 → ( ( ( 𝜓 ↔ 𝜃 ) ↔ 𝜂 ) ↔ ( ( 𝜒 ↔ 𝜏 ) ↔ 𝜁 ) ) ) |
6 | wl-3xorbi2 | ⊢ ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ ( ( 𝜓 ↔ 𝜃 ) ↔ 𝜂 ) ) | |
7 | wl-3xorbi2 | ⊢ ( hadd ( 𝜒 , 𝜏 , 𝜁 ) ↔ ( ( 𝜒 ↔ 𝜏 ) ↔ 𝜁 ) ) | |
8 | 5 6 7 | 3bitr4g | ⊢ ( 𝜑 → ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) ) ) |