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Theorem syl3an9b 1216
Description: Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
Hypotheses
Ref Expression
syl3an9b.1 (𝜑 → (𝜓𝜒))
syl3an9b.2 (𝜃 → (𝜒𝜏))
syl3an9b.3 (𝜂 → (𝜏𝜁))
Assertion
Ref Expression
syl3an9b ((𝜑𝜃𝜂) → (𝜓𝜁))

Proof of Theorem syl3an9b
StepHypRef Expression
1 syl3an9b.1 . . . 4 (𝜑 → (𝜓𝜒))
2 syl3an9b.2 . . . 4 (𝜃 → (𝜒𝜏))
31, 2sylan9bb 443 . . 3 ((𝜑𝜃) → (𝜓𝜏))
4 syl3an9b.3 . . 3 (𝜂 → (𝜏𝜁))
53, 4sylan9bb 443 . 2 (((𝜑𝜃) ∧ 𝜂) → (𝜓𝜁))
653impa 1110 1 ((𝜑𝜃𝜂) → (𝜓𝜁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  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:  eloprabg  5619
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