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Theorem sylan9 395
Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
Hypotheses
Ref Expression
sylan9.1 (𝜑 → (𝜓𝜒))
sylan9.2 (𝜃 → (𝜒𝜏))
Assertion
Ref Expression
sylan9 ((𝜑𝜃) → (𝜓𝜏))

Proof of Theorem sylan9
StepHypRef Expression
1 sylan9.1 . . 3 (𝜑 → (𝜓𝜒))
2 sylan9.2 . . 3 (𝜃 → (𝜒𝜏))
31, 2syl9 70 . 2 (𝜑 → (𝜃 → (𝜓𝜏)))
43imp 119 1 ((𝜑𝜃) → (𝜓𝜏))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104
This theorem is referenced by:  sbequi  1736  rspc2  2683  rspc3v  2688  trintssmOLD  3899  copsexg  4009  chfnrn  5306  ffnfv  5351  f1elima  5440  smoel2  5949  th3q  6242  addnnnq0  6605  mulnnnq0  6606  addsrpr  6888  mulsrpr  6889  cau3lem  9941
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