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Theorem syl6an 1339
Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
Hypotheses
Ref Expression
syl6an.1 (𝜑𝜓)
syl6an.2 (𝜑 → (𝜒𝜃))
syl6an.3 ((𝜓𝜃) → 𝜏)
Assertion
Ref Expression
syl6an (𝜑 → (𝜒𝜏))

Proof of Theorem syl6an
StepHypRef Expression
1 syl6an.2 . . 3 (𝜑 → (𝜒𝜃))
2 syl6an.1 . . 3 (𝜑𝜓)
31, 2jctild 303 . 2 (𝜑 → (𝜒 → (𝜓𝜃)))
4 syl6an.3 . 2 ((𝜓𝜃) → 𝜏)
53, 4syl6 33 1 (𝜑 → (𝜒𝜏))
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-ia3 105
This theorem is referenced by:  rdgon  6003  prarloclem5  6655  ltsopr  6751  nominpos  8218  ublbneg  8644  absle  9915  climshftlemg  10053  serif0  10101  dvds1lem  10118  dvds2lem  10119  bj-indind  10429
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