Theorem mp3and

Description: A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Hypotheses	mp3and.1	$$⊢\phi \to \psi$$
	mp3and.2	$$⊢\phi \to \chi$$
	mp3and.3	$$⊢\phi \to \theta$$
	mp3and.4	$$⊢\phi \to \left(\left(\psi \wedge \chi \wedge \theta \right)\to \tau \right)$$
Assertion	mp3and	$$⊢\phi \to \tau$$

Proof

Step	Hyp	Ref	Expression
1		mp3and.1	$$⊢\phi \to \psi$$
2		mp3and.2	$$⊢\phi \to \chi$$
3		mp3and.3	$$⊢\phi \to \theta$$
4		mp3and.4	$$⊢\phi \to \left(\left(\psi \wedge \chi \wedge \theta \right)\to \tau \right)$$
5	1 2 3	3jca	$$⊢\phi \to \left(\psi \wedge \chi \wedge \theta \right)$$
6	5 4	mpd	$$⊢\phi \to \tau$$