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Metamath Proof Explorer

Theorem syl6

Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993) (Proof shortened by Wolf Lammen, 30-Jul-2012)

	Ref	Expression
Hypotheses	syl6.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	syl6.2	⊢ ( 𝜒 → 𝜃 )
Assertion	syl6	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		syl6.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		syl6.2	⊢ ( 𝜒 → 𝜃 )
3	2	a1i	⊢ ( 𝜓 → ( 𝜒 → 𝜃 ) )
4	1 3	sylcom	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )