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Theorem 3bitr3d 216
Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
Hypotheses
Ref Expression
3bitr3d.1  |-  ( ph  ->  ( ps  <->  ch )
)
3bitr3d.2  |-  ( ph  ->  ( ps  <->  th )
)
3bitr3d.3  |-  ( ph  ->  ( ch  <->  ta )
)
Assertion
Ref Expression
3bitr3d  |-  ( ph  ->  ( th  <->  ta )
)

Proof of Theorem 3bitr3d
StepHypRef Expression
1 3bitr3d.2 . . 3  |-  ( ph  ->  ( ps  <->  th )
)
2 3bitr3d.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2bitr3d 188 . 2  |-  ( ph  ->  ( th  <->  ch )
)
4 3bitr3d.3 . 2  |-  ( ph  ->  ( ch  <->  ta )
)
53, 4bitrd 186 1  |-  ( ph  ->  ( th  <->  ta )
)
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:  csbcomg  2942  eloprabga  5673  ereldm  6268  mapen  6495  ordiso2  6649  subcan  7658  conjmulap  8112  ltrec  8256  divelunit  9328  fseq1m1p1  9416  fzm1  9421  fihashneq0  10052  hashfacen  10089  cvg1nlemcau  10258  lenegsq  10369  dvdsmod  10657  bezoutlemle  10791  rpexp  10926  qnumdenbi  10964
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