Theorem sylancom

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 2-Jul-2008)

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

Proof

Step	Hyp	Ref	Expression
1		sylancom.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		sylancom.2	⊢ ( ( 𝜒 ∧ 𝜓 ) → 𝜃 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜓 )
4	1 3 2	syl2anc	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )