Theorem sylc

Description: A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994) (Revised by NM, 13-Jul-2013)

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

Proof

Step	Hyp	Ref	Expression
1		sylc.1	⊢ ( 𝜑 → 𝜓 )
2		sylc.2	⊢ ( 𝜑 → 𝜒 )
3		sylc.3	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
4	1 2 3	syl2im	⊢ ( 𝜑 → ( 𝜑 → 𝜃 ) )
5	4	pm2.43i	⊢ ( 𝜑 → 𝜃 )