Theorem 3orbi123i

Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994)

	Ref	Expression
Hypotheses	bi3.1	⊢ ( 𝜑 ↔ 𝜓 )
	bi3.2	⊢ ( 𝜒 ↔ 𝜃 )
	bi3.3	⊢ ( 𝜏 ↔ 𝜂 )
Assertion	3orbi123i	⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜏 ) ↔ ( 𝜓 ∨ 𝜃 ∨ 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		bi3.1	⊢ ( 𝜑 ↔ 𝜓 )
2		bi3.2	⊢ ( 𝜒 ↔ 𝜃 )
3		bi3.3	⊢ ( 𝜏 ↔ 𝜂 )
4	1 2	orbi12i	⊢ ( ( 𝜑 ∨ 𝜒 ) ↔ ( 𝜓 ∨ 𝜃 ) )
5	4 3	orbi12i	⊢ ( ( ( 𝜑 ∨ 𝜒 ) ∨ 𝜏 ) ↔ ( ( 𝜓 ∨ 𝜃 ) ∨ 𝜂 ) )
6		df-3or	⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜏 ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∨ 𝜏 ) )
7		df-3or	⊢ ( ( 𝜓 ∨ 𝜃 ∨ 𝜂 ) ↔ ( ( 𝜓 ∨ 𝜃 ) ∨ 𝜂 ) )
8	5 6 7	3bitr4i	⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜏 ) ↔ ( 𝜓 ∨ 𝜃 ∨ 𝜂 ) )