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Theorem syl6bir 162
Description: A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
Hypotheses
Ref Expression
syl6bir.1  |-  ( ph  ->  ( ch  <->  ps )
)
syl6bir.2  |-  ( ch 
->  th )
Assertion
Ref Expression
syl6bir  |-  ( ph  ->  ( ps  ->  th )
)

Proof of Theorem syl6bir
StepHypRef Expression
1 syl6bir.1 . . 3  |-  ( ph  ->  ( ch  <->  ps )
)
21biimprd 156 . 2  |-  ( ph  ->  ( ps  ->  ch ) )
3 syl6bir.2 . 2  |-  ( ch 
->  th )
42, 3syl6 33 1  |-  ( ph  ->  ( ps  ->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  exdistrfor  1722  cbvexdh  1843  repizf2  3944  issref  4737  fnun  5036  ovigg  5652  tfrlem9  5968  tfri3  6016  ordge1n0im  6083  nntri3or  6137  axprecex  7108  peano5nnnn  7120  peano5nni  8109  zeo  8533  nn0ind-raph  8545  fzm1  9193  fzind2  9325  fzfig  9512  bcpasc  9790  climrecvg1n  10323  oddnn02np1  10424  oddge22np1  10425  evennn02n  10426  evennn2n  10427  gcdaddm  10519  coprmdvds1  10617  qredeq  10622  bj-intabssel  10750
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