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  6204  sbcoteq1a  8009  fnwelem  8087  mpocurryd  8225  compssiso  10303  divfl0  13762  cjreb  15065  cnpart  15182  bitsuz  16420  acsfn  17596  eqg0el  19091  ghmeqker  19151  odmulg  19462  psrbaglefi  21811  cnrest2  23149  hausdiag  23508  prdsbl  24355  mcubic  26733  2lgslem1a2  27277  fmptco1f1o  32530  qsidomlem2  33397  areacirclem4  37678  lmclim2  37725  cmtbr2N  39219  expdiophlem1  42983  cantnfresb  43286  rrx2linest  48704