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Theorem syl3an2 1180
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)
Hypotheses
Ref Expression
syl3an2.1 (𝜑𝜒)
syl3an2.2 ((𝜓𝜒𝜃) → 𝜏)
Assertion
Ref Expression
syl3an2 ((𝜓𝜑𝜃) → 𝜏)

Proof of Theorem syl3an2
StepHypRef Expression
1 syl3an2.1 . . 3 (𝜑𝜒)
2 syl3an2.2 . . . 4 ((𝜓𝜒𝜃) → 𝜏)
323exp 1114 . . 3 (𝜓 → (𝜒 → (𝜃𝜏)))
41, 3syl5 32 . 2 (𝜓 → (𝜑 → (𝜃𝜏)))
543imp 1109 1 ((𝜓𝜑𝜃) → 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by:  syl3an2b  1183  syl3an2br  1186  syl3anl2  1195  nndi  6095  nnmass  6096  prarloclemarch2  6574  1idprl  6745  1idpru  6746  recexprlem1ssl  6788  recexprlem1ssu  6789  msqge0  7680  mulge0  7683  divsubdirap  7758  divdiv32ap  7770  peano2uz  8621  fzoshftral  9195  expdivap  9470  ibcval5  9630  redivap  9701  imdivap  9708  absdiflt  9918  absdifle  9919
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