Theorem 3anbi3i

Description: Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006)

	Ref	Expression
Hypothesis	3anbi1i.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	3anbi3i	⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		3anbi1i.1	⊢ ( 𝜑 ↔ 𝜓 )
2		biid	⊢ ( 𝜒 ↔ 𝜒 )
3		biid	⊢ ( 𝜃 ↔ 𝜃 )
4	2 3 1	3anbi123i	⊢ ( ( 𝜒 ∧ 𝜃 ∧ 𝜑 ) ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜓 ) )