Theorem syl6an

Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011)

	Ref	Expression
Hypotheses	syl6an.1	⊢ ( 𝜑 → 𝜓 )
	syl6an.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
	syl6an.3	⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
Assertion	syl6an	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6an.1	⊢ ( 𝜑 → 𝜓 )
2		syl6an.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
3		syl6an.3	⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜏 )
4	3	ex	⊢ ( 𝜓 → ( 𝜃 → 𝜏 ) )
5	1 2 4	sylsyld	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )