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  10484  fzosplitsnm1  10638  fzofzp1  10656  fzostep1  10667  bcm1k  11214  pfxccatpfx2  11525  climuni  12078  serf0  12137  fsumparts  12256  hashiun  12264  oddprm  13061  znzrh2  15065  znf1o  15070  znidom  15076  hmeores  15507  ppinprm  16221  chtnprm  16223  gausslemma2dlem0c  16336  gausslemma2dlem0e  16338  gausslemma2dlem1a  16343  eupthvdres  16882