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Theorem syl6mpi 64
Description: A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.)
Hypotheses
Ref Expression
syl6mpi.1 (𝜑 → (𝜓𝜒))
syl6mpi.2 𝜃
syl6mpi.3 (𝜒 → (𝜃𝜏))
Assertion
Ref Expression
syl6mpi (𝜑 → (𝜓𝜏))

Proof of Theorem syl6mpi
StepHypRef Expression
1 syl6mpi.1 . 2 (𝜑 → (𝜓𝜒))
2 syl6mpi.2 . . 3 𝜃
3 syl6mpi.3 . . 3 (𝜒 → (𝜃𝜏))
42, 3mpi 20 . 2 (𝜒𝜏)
51, 4syl6 34 1 (𝜑 → (𝜓𝜏))
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:  19.8a  2038  19.8aOLD  2039  suceloni  6882  bndrank  8564  ac10ct  8717  1re  9895  tratrb  37563  ee20an  37773  uspgrn2crct  41006
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