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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	\|- ( ph -> ps )
	4syl.2	\|- ( ps -> ch )
	4syl.3	\|- ( ch -> th )
	4syl.4	\|- ( th -> ta )
Assertion	4syl	\|- ( ph -> ta )

Proof

Step	Hyp	Ref	Expression
1		4syl.1	\|- ( ph -> ps )
2		4syl.2	\|- ( ps -> ch )
3		4syl.3	\|- ( ch -> th )
4		4syl.4	\|- ( th -> ta )
5	1 2 3	3syl	\|- ( ph -> th )
6	5 4	syl	\|- ( ph -> ta )