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	⊢ (𝜑 → 𝜓)
4syl.2	⊢ (𝜓 → 𝜒)
4syl.3	⊢ (𝜒 → 𝜃)
4syl.4	⊢ (𝜃 → 𝜏)

Assertion

Ref	Expression
4syl	⊢ (𝜑 → 𝜏)

Proof of Theorem 4syl

Step	Hyp	Ref	Expression
1		4syl.1	. . 3 ⊢ (𝜑 → 𝜓)
2		4syl.2	. . 3 ⊢ (𝜓 → 𝜒)
3		4syl.3	. . 3 ⊢ (𝜒 → 𝜃)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝜑 → 𝜃)
5		4syl.4	. 2 ⊢ (𝜃 → 𝜏)
6	4, 5	syl 14	1 ⊢ (𝜑 → 𝜏)

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  5961  fcof1o  5968  isoselem  5999  isose  6000  tposss  6490  smoiso  6546  fzssp1  10422  fzosplitsnm1  10576  fzofzp1  10594  fzostep1  10605  bcm1k  11147  pfxccatpfx2  11454  climuni  12003  serf0  12062  fsumparts  12181  hashiun  12189  oddprm  12982  znzrh2  14906  znf1o  14911  znidom  14917  hmeores  15292  gausslemma2dlem0c  16036  gausslemma2dlem0e  16038  gausslemma2dlem1a  16043  eupthvdres  16582