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Theorem syldanl 730
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 448 . . 3 (𝜑 → (𝜓𝜒))
32imdistani 721 . 2 ((𝜑𝜓) → (𝜑𝜒))
4 syldanl.2 . 2 (((𝜑𝜒) ∧ 𝜃) → 𝜏)
53, 4sylan 486 1 (((𝜑𝜓) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382
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 195  df-an 384
This theorem is referenced by:  trust  21785  submateq  29009  heibor1lem  32574  idlnegcl  32787  igenmin  32829  binomcxplemnotnn0  37373  vonioolem1  39368  vonicclem1  39371
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