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-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  f1ocnvfvrneq  5933  fcof1o  5940  isoselem  5971  isose  5972  tposss  6455  smoiso  6511  fzssp1  10364  fzosplitsnm1  10517  fzofzp1  10535  fzostep1  10546  bcm1k  11085  pfxccatpfx2  11384  climuni  11933  serf0  11992  fsumparts  12111  hashiun  12119  oddprm  12912  znzrh2  14742  znf1o  14747  znidom  14753  hmeores  15126  gausslemma2dlem0c  15870  gausslemma2dlem0e  15872  gausslemma2dlem1a  15877  eupthvdres  16416