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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 (𝜑𝜓)
4syl.2 (𝜓𝜒)
4syl.3 (𝜒𝜃)
4syl.4 (𝜃𝜏)
Assertion
Ref Expression
4syl (𝜑𝜏)

Proof of Theorem 4syl
StepHypRef Expression
1 4syl.1 . . 3 (𝜑𝜓)
2 4syl.2 . . 3 (𝜓𝜒)
3 4syl.3 . . 3 (𝜒𝜃)
41, 2, 33syl 17 . 2 (𝜑𝜃)
5 4syl.4 . 2 (𝜃𝜏)
64, 5syl 14 1 (𝜑𝜏)
Colors of variables: wff set class
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  5651  fcof1o  5658  isoselem  5689  isose  5690  tposss  6111  smoiso  6167  fzssp1  9815  fzosplitsnm1  9954  fzofzp1  9972  fzostep1  9982  bcm1k  10474  climuni  11030  serf0  11089  fsumparts  11207  hashiun  11215  hmeores  12411
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