Theorem anim12ii

Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007) (Proof shortened by Wolf Lammen, 19-Jul-2013)

	Ref	Expression
Hypotheses	anim12ii.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	anim12ii.2	⊢ ( 𝜃 → ( 𝜓 → 𝜏 ) )
Assertion	anim12ii	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → ( 𝜒 ∧ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		anim12ii.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		anim12ii.2	⊢ ( 𝜃 → ( 𝜓 → 𝜏 ) )
3		pm3.43	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜓 → 𝜏 ) ) → ( 𝜓 → ( 𝜒 ∧ 𝜏 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜓 → ( 𝜒 ∧ 𝜏 ) ) )