Theorem 3anbi1d

Description: Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006)

	Ref	Expression
Hypothesis	3anbi1d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	3anbi1d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anbi1d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		biidd	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜃 ) )
3	1 2	3anbi12d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ∧ 𝜏 ) ↔ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) )