Theorem bi2anan9

Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995)

	Ref	Expression
Hypotheses	bi2an9.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	bi2an9.2	⊢ ( 𝜃 → ( 𝜏 ↔ 𝜂 ) )
Assertion	bi2anan9	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) ↔ ( 𝜒 ∧ 𝜂 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi2an9.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2		bi2an9.2	⊢ ( 𝜃 → ( 𝜏 ↔ 𝜂 ) )
3		pm4.38	⊢ ( ( ( 𝜓 ↔ 𝜒 ) ∧ ( 𝜏 ↔ 𝜂 ) ) → ( ( 𝜓 ∧ 𝜏 ) ↔ ( 𝜒 ∧ 𝜂 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) ↔ ( 𝜒 ∧ 𝜂 ) ) )