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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	$⊢ φ \to (ψ \to χ)$
	syl6.2	$⊢ χ \to θ$
Assertion	syl6	$⊢ φ \to (ψ \to θ)$

Proof

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