Theorem mpand

Description: A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004) (Proof shortened by Wolf Lammen, 7-Apr-2013)

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

Proof

Step	Hyp	Ref	Expression
1		mpand.1	⊢ ( 𝜑 → 𝜓 )
2		mpand.2	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
3	2	ancomsd	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝜓 ) → 𝜃 ) )
4	1 3	mpan2d	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )