Theorem mpsylsyld

Description: Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012)

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

Proof

Step	Hyp	Ref	Expression
1		mpsylsyld.1	⊢ 𝜑
2		mpsylsyld.2	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
3		mpsylsyld.3	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
4	1	a1i	⊢ ( 𝜓 → 𝜑 )
5	4 2 3	sylsyld	⊢ ( 𝜓 → ( 𝜒 → 𝜏 ) )