Theorem 4syl 18

Description: Inference chaining three syllogisms. The use of this theorem is marked "discouraged" because it can cause the "minimize" command to have very long run times. However, feel free to use "minimize 4syl /override" if you wish. (Contributed by BJ, 14-Jul-2018.) (New usage is discouraged.)

Hypotheses

Ref	Expression
4syl.1	|- ( ph -> ps )
4syl.2	|- ( ps -> ch )
4syl.3	|- ( ch -> th )
4syl.4	|- ( th -> ta )

Assertion

Ref	Expression
4syl	|- ( ph -> ta )

Proof of Theorem 4syl

Step	Hyp	Ref	Expression
1		4syl.1	. . 3 |- ( ph -> ps )
2		4syl.2	. . 3 |- ( ps -> ch )
3		4syl.3	. . 3 |- ( ch -> th )
4	1, 2, 3	3syl 17	. 2 |- ( ph -> th )
5		4syl.4	. 2 |- ( th -> ta )
6	4, 5	syl 14	1 |- ( ph -> ta )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7

This theorem is referenced by:  f1ocnvfvrneq  5422  fcof1o  5429  isoselem  5459  isose  5460  tposss  5861  smoiso  5917  fzssp1  8930  fzosplitsnm1  9065  fzofzp1  9083  fzostep1  9093  climuni  9814  serif0  9871