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Theorem syl3an3 1181
Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)
Hypotheses
Ref Expression
syl3an3.1 (𝜑𝜃)
syl3an3.2 ((𝜓𝜒𝜃) → 𝜏)
Assertion
Ref Expression
syl3an3 ((𝜓𝜒𝜑) → 𝜏)

Proof of Theorem syl3an3
StepHypRef Expression
1 syl3an3.1 . . 3 (𝜑𝜃)
2 syl3an3.2 . . . 4 ((𝜓𝜒𝜃) → 𝜏)
323exp 1114 . . 3 (𝜓 → (𝜒 → (𝜃𝜏)))
41, 3syl7 67 . 2 (𝜓 → (𝜒 → (𝜑𝜏)))
543imp 1109 1 ((𝜓𝜒𝜑) → 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:  syl3an3b  1184  syl3an3br  1187  vtoclgft  2621  ovmpt2x  5657  ovmpt2ga  5658  nnanq0  6614  apreim  7668  divassap  7743  ltmul2  7897  elfzo  9108  subcn2  10063  mulcn2  10064  ndvdsp1  10244
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