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  5954  fcof1o  5961  isoselem  5992  isose  5993  tposss  6476  smoiso  6532  fzssp1  10400  fzosplitsnm1  10553  fzofzp1  10571  fzostep1  10582  bcm1k  11121  pfxccatpfx2  11425  climuni  11974  serf0  12033  fsumparts  12152  hashiun  12160  oddprm  12953  znzrh2  14786  znf1o  14791  znidom  14797  hmeores  15172  gausslemma2dlem0c  15916  gausslemma2dlem0e  15918  gausslemma2dlem1a  15923  eupthvdres  16462