Theorem 3anbi123i

Description: Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994)

	Ref	Expression
Hypotheses	bi3.1	⊢ ( 𝜑 ↔ 𝜓 )
	bi3.2	⊢ ( 𝜒 ↔ 𝜃 )
	bi3.3	⊢ ( 𝜏 ↔ 𝜂 )
Assertion	3anbi123i	⊢ ( ( 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	anbi12i	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) )
5	4 3	anbi12i	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜏 ) ↔ ( ( 𝜓 ∧ 𝜃 ) ∧ 𝜂 ) )
6		df-3an	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ 𝜏 ) )
7		df-3an	⊢ ( ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) ↔ ( ( 𝜓 ∧ 𝜃 ) ∧ 𝜂 ) )
8	5 6 7	3bitr4i	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) ↔ ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) )