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Theorem syl6an 1427
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 314 . 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 103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 107
This theorem is referenced by:  mapxpen  6826  prarloclem5  7462  ltsopr  7558  suplocsrlem  7770  nominpos  9115  ublbneg  9572  absle  11053  rexanre  11184  rexico  11185  climshftlemg  11265  serf0  11315  dvds1lem  11764  dvds2lem  11765  lmconst  13010  addcncntoplem  13345  bj-indind  13967
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