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Theorem syldanl 595
Description: A syllogism deduction with conjoined antecedents. (Contributed by Jeff Madsen, 20-Jun-2011.)
Hypotheses
Ref Expression
syldanl.1 ((𝜑𝜓) → 𝜒)
syldanl.2 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
Assertion
Ref Expression
syldanl (((𝜑𝜓) ∧ 𝜃) → 𝜏)

Proof of Theorem syldanl
StepHypRef Expression
1 syldanl.1 . . . 4 ((𝜑𝜓) → 𝜒)
21ex 403 . . 3 (𝜑 → (𝜓𝜒))
32imdistani 564 . 2 ((𝜑𝜓) → (𝜑𝜒))
4 syldanl.2 . 2 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
53, 4sylan 575 1 (((𝜑𝜓) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-an 387
This theorem is referenced by:  oen0  7950  oeordsuc  7958  erth  8073  lo1bdd2  14663  grplmulf1o  17876  grplactcnv  17905  trust  22441  submateq  30473  heibor1lem  34232  idlnegcl  34445  igenmin  34487  eqvrelth  34981  binomcxplemnotnn0  39511  vonioolem1  41821  vonicclem1  41824  smfsuplem1  41944  smflimsuplem4  41956
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