Theorem 3anbi23d

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006)

	Ref	Expression
Hypotheses	3anbi12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	3anbi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion	3anbi23d	⊢ ( 𝜑 → ( ( 𝜂 ∧ 𝜓 ∧ 𝜃 ) ↔ ( 𝜂 ∧ 𝜒 ∧ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anbi12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		3anbi12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
3		biidd	⊢ ( 𝜑 → ( 𝜂 ↔ 𝜂 ) )
4	3 1 2	3anbi123d	⊢ ( 𝜑 → ( ( 𝜂 ∧ 𝜓 ∧ 𝜃 ) ↔ ( 𝜂 ∧ 𝜒 ∧ 𝜏 ) ) )