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Metamath Proof Explorer

Theorem syl7

Description: A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993) (Proof shortened by Wolf Lammen, 3-Aug-2012)

	Ref	Expression
Hypotheses	syl7.1	⊢ ( 𝜑 → 𝜓 )
	syl7.2	⊢ ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )
Assertion	syl7	⊢ ( 𝜒 → ( 𝜃 → ( 𝜑 → 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		syl7.1	⊢ ( 𝜑 → 𝜓 )
2		syl7.2	⊢ ( 𝜒 → ( 𝜃 → ( 𝜓 → 𝜏 ) ) )
3	1	a1i	⊢ ( 𝜒 → ( 𝜑 → 𝜓 ) )
4	3 2	syl5d	⊢ ( 𝜒 → ( 𝜃 → ( 𝜑 → 𝜏 ) ) )