Theorem wl-3xorbi123i

Description: Equivalence theorem for triple xor. Copy of hadbi123i . (Contributed by Mario Carneiro, 4-Sep-2016)

	Ref	Expression
Hypotheses	wl-3xorbii.1	⊢ ( 𝜓 ↔ 𝜒 )
	wl-3xorbii.2	⊢ ( 𝜃 ↔ 𝜏 )
	wl-3xorbii.3	⊢ ( 𝜂 ↔ 𝜁 )
Assertion	wl-3xorbi123i	⊢ ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) )

Proof

Step	Hyp	Ref	Expression
1		wl-3xorbii.1	⊢ ( 𝜓 ↔ 𝜒 )
2		wl-3xorbii.2	⊢ ( 𝜃 ↔ 𝜏 )
3		wl-3xorbii.3	⊢ ( 𝜂 ↔ 𝜁 )
4	1	a1i	⊢ ( ⊤ → ( 𝜓 ↔ 𝜒 ) )
5	2	a1i	⊢ ( ⊤ → ( 𝜃 ↔ 𝜏 ) )
6	3	a1i	⊢ ( ⊤ → ( 𝜂 ↔ 𝜁 ) )
7	4 5 6	wl-3xorbi123d	⊢ ( ⊤ → ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) ) )
8	7	mptru	⊢ ( hadd ( 𝜓 , 𝜃 , 𝜂 ) ↔ hadd ( 𝜒 , 𝜏 , 𝜁 ) )