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)

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  5991  fcof1o  5998  isoselem  6029  isose  6030  tposss  6522  smoiso  6578  fzssp1  10490  fzosplitsnm1  10644  fzofzp1  10662  fzostep1  10673  bcm1k  11220  pfxccatpfx2  11531  climuni  12084  serf0  12143  fsumparts  12262  hashiun  12270  oddprm  13067  znzrh2  15090  znf1o  15095  znidom  15101  hmeores  15574  ppinprm  16293  chtnprm  16295  gausslemma2dlem0c  16408  gausslemma2dlem0e  16410  gausslemma2dlem1a  16415  eupthvdres  16954