Theorem mp3and

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

	Ref	Expression
Hypotheses	mp3and.1	⊢ ( 𝜑 → 𝜓 )
	mp3and.2	⊢ ( 𝜑 → 𝜒 )
	mp3and.3	⊢ ( 𝜑 → 𝜃 )
	mp3and.4	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 ) )
Assertion	mp3and	⊢ ( 𝜑 → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		mp3and.1	⊢ ( 𝜑 → 𝜓 )
2		mp3and.2	⊢ ( 𝜑 → 𝜒 )
3		mp3and.3	⊢ ( 𝜑 → 𝜃 )
4		mp3and.4	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 ) )
5	1 2 3	3jca	⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
6	5 4	mpd	⊢ ( 𝜑 → 𝜏 )