Theorem syl2anr

Description: A double syllogism inference. For an implication-only version, see syl2imc . (Contributed by NM, 17-Sep-2013)

	Ref	Expression
Hypotheses	syl2an.1	⊢ ( 𝜑 → 𝜓 )
	syl2an.2	⊢ ( 𝜏 → 𝜒 )
	syl2an.3	⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion	syl2anr	⊢ ( ( 𝜏 ∧ 𝜑 ) → 𝜃 )

Proof

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