ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3imtr3d Unicode version

Theorem 3imtr3d 191
Description: More general version of 3imtr3i 189. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
Hypotheses
Ref Expression
3imtr3d.1  |-  ( ph  ->  ( ps  ->  ch ) )
3imtr3d.2  |-  ( ph  ->  ( ps  <->  th )
)
3imtr3d.3  |-  ( ph  ->  ( ch  <->  ta )
)
Assertion
Ref Expression
3imtr3d  |-  ( ph  ->  ( th  ->  ta ) )

Proof of Theorem 3imtr3d
StepHypRef Expression
1 3imtr3d.2 . 2  |-  ( ph  ->  ( ps  <->  th )
)
2 3imtr3d.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
3 3imtr3d.3 . . 3  |-  ( ph  ->  ( ch  <->  ta )
)
42, 3sylibd 138 . 2  |-  ( ph  ->  ( ps  ->  ta ) )
51, 4sylbird 159 1  |-  ( ph  ->  ( th  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  f1imass  5413  fornex  5742  tposfn2  5881  freccl  5993  eroveu  6197  indpi  6440  axcaucvglemres  6973  caucvgrelemcau  9579
  Copyright terms: Public domain W3C validator