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Theorem 3bitrrd 295
Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
Hypotheses
Ref Expression
3bitrd.1 (𝜑 → (𝜓𝜒))
3bitrd.2 (𝜑 → (𝜒𝜃))
3bitrd.3 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
3bitrrd (𝜑 → (𝜏𝜓))

Proof of Theorem 3bitrrd
StepHypRef Expression
1 3bitrd.3 . 2 (𝜑 → (𝜃𝜏))
2 3bitrd.1 . . 3 (𝜑 → (𝜓𝜒))
3 3bitrd.2 . . 3 (𝜑 → (𝜒𝜃))
42, 3bitr2d 269 . 2 (𝜑 → (𝜃𝜓))
51, 4bitr3d 270 1 (𝜑 → (𝜏𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197
This theorem is referenced by:  fnwelem  7337  mpt2curryd  7440  compssiso  9234  divfl0  12665  cjreb  13907  cnpart  14024  bitsuz  15243  acsfn  16367  ghmeqker  17734  odmulg  18019  psrbaglefi  19420  cnrest2  21138  hausdiag  21496  prdsbl  22343  mcubic  24619  2lgslem1a2  25160  fmptco1f1o  29562  areacirclem4  33633  lmclim2  33684  cmtbr2N  34858  expdiophlem1  37905  rnmpt0  39726
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