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Theorem syl10 79
Description: A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
Hypotheses
Ref Expression
syl10.1 (𝜑 → (𝜓𝜒))
syl10.2 (𝜑 → (𝜓 → (𝜃𝜏)))
syl10.3 (𝜒 → (𝜏𝜂))
Assertion
Ref Expression
syl10 (𝜑 → (𝜓 → (𝜃𝜂)))

Proof of Theorem syl10
StepHypRef Expression
1 syl10.2 . 2 (𝜑 → (𝜓 → (𝜃𝜏)))
2 syl10.1 . . 3 (𝜑 → (𝜓𝜒))
3 syl10.3 . . 3 (𝜒 → (𝜏𝜂))
42, 3syl6 35 . 2 (𝜑 → (𝜓 → (𝜏𝜂)))
51, 4syldd 72 1 (𝜑 → (𝜓 → (𝜃𝜂)))
Colors of variables: wff setvar 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:  tz7.49  7485  rspsbc2  38226  tratrb  38228
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