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 )

This proof depends on syntax axioms:   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  f1ocnvfvrneq  5988  fcof1o  5995  isoselem  6026  isose  6027  tposss  6517  smoiso  6573  fzssp1  10483  fzosplitsnm1  10637  fzofzp1  10655  fzostep1  10666  bcm1k  11212  pfxccatpfx2  11523  climuni  12075  serf0  12134  fsumparts  12253  hashiun  12261  oddprm  13058  znzrh2  15030  znf1o  15035  znidom  15041  hmeores  15465  ppinprm  16171  gausslemma2dlem0c  16268  gausslemma2dlem0e  16270  gausslemma2dlem1a  16275  eupthvdres  16814