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  5750  fcof1o  5757  isoselem  5788  isose  5789  tposss  6214  smoiso  6270  fzssp1  10002  fzosplitsnm1  10144  fzofzp1  10162  fzostep1  10172  bcm1k  10673  climuni  11234  serf0  11293  fsumparts  11411  hashiun  11419  oddprm  12191  hmeores  12955