Theorem im2anan9

Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996)

	Ref	Expression
Hypotheses	im2an9.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	im2an9.2	⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) )
Assertion	im2anan9	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) → ( 𝜒 ∧ 𝜂 ) ) )

Proof

Step	Hyp	Ref	Expression
1		im2an9.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		im2an9.2	⊢ ( 𝜃 → ( 𝜏 → 𝜂 ) )
3	1	adantrd	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜏 ) → 𝜒 ) )
4	2	adantld	⊢ ( 𝜃 → ( ( 𝜓 ∧ 𝜏 ) → 𝜂 ) )
5	3 4	anim12ii	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜓 ∧ 𝜏 ) → ( 𝜒 ∧ 𝜂 ) ) )