Theorem sylancr

Description: Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009)

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

Proof

Step	Hyp	Ref	Expression
1		sylancr.1	⊢ 𝜓
2		sylancr.2	⊢ ( 𝜑 → 𝜒 )
3		sylancr.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4	1	a1i	⊢ ( 𝜑 → 𝜓 )
5	4 2 3	syl2anc	⊢ ( 𝜑 → 𝜃 )