Theorem syl2anc

Description: Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012)

	Ref	Expression
Hypotheses	syl2anc.1	⊢ ( 𝜑 → 𝜓 )
	syl2anc.2	⊢ ( 𝜑 → 𝜒 )
	syl2anc.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion	syl2anc	⊢ ( 𝜑 → 𝜃 )

Proof

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