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Metamath Proof Explorer

Theorem sylcom

Description: Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004) (Proof shortened by Mel L. O'Cat, 2-Feb-2006) (Proof shortened by Stefan Allan, 23-Feb-2006)

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

Proof

Step	Hyp	Ref	Expression
1		sylcom.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		sylcom.2	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
3	2	a2i	⊢ ( ( 𝜓 → 𝜒 ) → ( 𝜓 → 𝜃 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )