Theorem 3bitrrd 306

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3bitrd.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
3bitrd.2	⊢ (𝜑 → (𝜒 ↔ 𝜃))
3bitrd.3	⊢ (𝜑 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
3bitrrd	⊢ (𝜑 → (𝜏 ↔ 𝜓))

Proof of Theorem 3bitrrd

Step	Hyp	Ref	Expression
1		3bitrd.3	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
2		3bitrd.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3		3bitrd.2	. . 3 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
4	2, 3	bitr2d 280	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜓))
5	1, 4	bitr3d 281	1 ⊢ (𝜑 → (𝜏 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206

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 207

This theorem is referenced by:  rnmpt0f  6274  sbcoteq1a  8092  fnwelem  8172  mpocurryd  8310  compssiso  10443  divfl0  13875  cjreb  15172  cnpart  15289  bitsuz  16520  acsfn  17717  eqg0el  19223  ghmeqker  19283  odmulg  19598  psrbaglefi  21969  cnrest2  23315  hausdiag  23674  prdsbl  24525  mcubic  26908  2lgslem1a2  27452  fmptco1f1o  32652  qsidomlem2  33446  areacirclem4  37671  lmclim2  37718  cmtbr2N  39209  expdiophlem1  42978  cantnfresb  43286  rrx2linest  48476