Theorem mp2and

Description: A deduction based on modus ponens. (Contributed by NM, 12-Dec-2004)

	Ref	Expression
Hypotheses	mp2and.1	⊢ ( 𝜑 → 𝜓 )
	mp2and.2	⊢ ( 𝜑 → 𝜒 )
	mp2and.3	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
Assertion	mp2and	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		mp2and.1	⊢ ( 𝜑 → 𝜓 )
2		mp2and.2	⊢ ( 𝜑 → 𝜒 )
3		mp2and.3	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
4	1 3	mpand	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
5	2 4	mpd	⊢ ( 𝜑 → 𝜃 )