Theorem anbi12i

Description: Conjoin both sides of two equivalences. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypotheses	anbi12.1	⊢ ( 𝜑 ↔ 𝜓 )
	anbi12.2	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	anbi12i	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		anbi12.1	⊢ ( 𝜑 ↔ 𝜓 )
2		anbi12.2	⊢ ( 𝜒 ↔ 𝜃 )
3	1	anbi1i	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜒 ) )
4	2	anbi2i	⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) )
5	3 4	bitri	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜃 ) )