Theorem mp2and

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

	Ref	Expression
Hypotheses	mp2and.1	$$⊢\phi \to \psi$$
	mp2and.2	$$⊢\phi \to \chi$$
	mp2and.3	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \theta \right)$$
Assertion	mp2and	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		mp2and.1	$$⊢\phi \to \psi$$
2		mp2and.2	$$⊢\phi \to \chi$$
3		mp2and.3	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \theta \right)$$
4	1 3	mpand	$$⊢\phi \to \left(\chi \to \theta \right)$$
5	2 4	mpd	$$⊢\phi \to \theta$$