Theorem mpsyl

Description: Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011)

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

Proof

Step	Hyp	Ref	Expression
1		mpsyl.1	⊢ 𝜑
2		mpsyl.2	⊢ ( 𝜓 → 𝜒 )
3		mpsyl.3	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
4	1	a1i	⊢ ( 𝜓 → 𝜑 )
5	4 2 3	sylc	⊢ ( 𝜓 → 𝜃 )