Theorem syl2an

Description: A double syllogism inference. For an implication-only version, see syl2im . (Contributed by NM, 31-Jan-1997)

	Ref	Expression
Hypotheses	syl2an.1	⊢ ( 𝜑 → 𝜓 )
	syl2an.2	⊢ ( 𝜏 → 𝜒 )
	syl2an.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion	syl2an	⊢ ( ( 𝜑 ∧ 𝜏 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		syl2an.1	⊢ ( 𝜑 → 𝜓 )
2		syl2an.2	⊢ ( 𝜏 → 𝜒 )
3		syl2an.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4	1 3	sylan	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
5	2 4	sylan2	⊢ ( ( 𝜑 ∧ 𝜏 ) → 𝜃 )