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Metamath Proof Explorer

Theorem syl5

Description: A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992) (Proof shortened by Wolf Lammen, 25-May-2013)

	Ref	Expression
Hypotheses	syl5.1	⊢ ( 𝜑 → 𝜓 )
	syl5.2	⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
Assertion	syl5	⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		syl5.1	⊢ ( 𝜑 → 𝜓 )
2		syl5.2	⊢ ( 𝜒 → ( 𝜓 → 𝜃 ) )
3	1 2	syl5com	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
4	3	com12	⊢ ( 𝜒 → ( 𝜑 → 𝜃 ) )