Metamath Proof Explorer


Theorem wl-3xorbi123d

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 → ( 𝜓𝜒 ) )
wl-3xorbid.2 ( 𝜑 → ( 𝜃𝜏 ) )
wl-3xorbid.3 ( 𝜑 → ( 𝜂𝜁 ) )
Assertion wl-3xorbi123d ( 𝜑 → ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) ) )

Proof

Step Hyp Ref Expression
1 wl-3xorbid.1 ( 𝜑 → ( 𝜓𝜒 ) )
2 wl-3xorbid.2 ( 𝜑 → ( 𝜃𝜏 ) )
3 wl-3xorbid.3 ( 𝜑 → ( 𝜂𝜁 ) )
4 1 2 bibi12d ( 𝜑 → ( ( 𝜓𝜃 ) ↔ ( 𝜒𝜏 ) ) )
5 4 3 bibi12d ( 𝜑 → ( ( ( 𝜓𝜃 ) ↔ 𝜂 ) ↔ ( ( 𝜒𝜏 ) ↔ 𝜁 ) ) )
6 wl-3xorbi2 ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ ( ( 𝜓𝜃 ) ↔ 𝜂 ) )
7 wl-3xorbi2 ( hadd ( 𝜒 , 𝜏 , 𝜁 ) ↔ ( ( 𝜒𝜏 ) ↔ 𝜁 ) )
8 5 6 7 3bitr4g ( 𝜑 → ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) ) )