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Theorem 3imtr3d 201
Description: More general version of 3imtr3i 199. 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 148 . 2  |-  ( ph  ->  ( ps  ->  ta ) )
51, 4sylbird 169 1  |-  ( ph  ->  ( th  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  f1imass  5726  fornex  6065  tposfn2  6215  eroveu  6573  ismkvnex  7100  indpi  7264  axcaucvglemres  7821  caucvgrelemcau  10891  m1dvdsndvds  12138  pcpremul  12183  limccnpcntop  13114  sincosq1sgn  13217  sincosq2sgn  13218  subctctexmid  13644
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