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  10473  fzosplitsnm1  10627  fzofzp1  10645  fzostep1  10656  bcm1k  11198  pfxccatpfx2  11509  climuni  12059  serf0  12118  fsumparts  12237  hashiun  12245  oddprm  13038  znzrh2  14981  znf1o  14986  znidom  14992  hmeores  15416  gausslemma2dlem0c  16170  gausslemma2dlem0e  16172  gausslemma2dlem1a  16177  eupthvdres  16716