Theorem anim12dan

Description: Conjoin antecedents and consequents in a deduction. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Hypotheses	anim12dan.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	anim12dan.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
Assertion	anim12dan	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜃 ) ) → ( 𝜒 ∧ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		anim12dan.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		anim12dan.2	⊢ ( ( 𝜑 ∧ 𝜃 ) → 𝜏 )
3	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	2	ex	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
5	3 4	anim12d	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜃 ) → ( 𝜒 ∧ 𝜏 ) ) )
6	5	imp	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜃 ) ) → ( 𝜒 ∧ 𝜏 ) )