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  6216  sbcoteq1a  8030  fnwelem  8110  mpocurryd  8248  compssiso  10327  divfl0  13786  cjreb  15089  cnpart  15206  bitsuz  16444  acsfn  17620  eqg0el  19115  ghmeqker  19175  odmulg  19486  psrbaglefi  21835  cnrest2  23173  hausdiag  23532  prdsbl  24379  mcubic  26757  2lgslem1a2  27301  fmptco1f1o  32557  qsidomlem2  33424  areacirclem4  37705  lmclim2  37752  cmtbr2N  39246  expdiophlem1  43010  cantnfresb  43313  rrx2linest  48731