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Metamath Proof Explorer

Theorem 4syl

Description: Inference chaining three syllogisms syl . (Contributed by BJ, 14-Jul-2018) The use of this theorem is marked "discouraged" because it can cause the Metamath program "MM-PA> MINIMIZE__WITH *" command to have very long run times. However, feel free to use "MM-PA> MINIMIZE__WITH 4syl / OVERRIDE" if you wish. Remember to update the "discouraged" file if it gets used. (New usage is discouraged.)

	Ref	Expression
Hypotheses	4syl.1	$⊢ φ \to ψ$
	4syl.2	$⊢ ψ \to χ$
	4syl.3	$⊢ χ \to θ$
	4syl.4	$⊢ θ \to τ$
Assertion	4syl	$⊢ φ \to τ$

Proof

Step	Hyp	Ref	Expression
1		4syl.1	$⊢ φ \to ψ$
2		4syl.2	$⊢ ψ \to χ$
3		4syl.3	$⊢ χ \to θ$
4		4syl.4	$⊢ θ \to τ$
5	1 2 3	3syl	$⊢ φ \to θ$
6	5 4	syl	$⊢ φ \to τ$