Theorem bitr3di 195

Description: A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Hypotheses

Ref	Expression
bitr3di.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bitr3di.2	⊢ (𝜓 ↔ 𝜃)

Assertion

Ref	Expression
bitr3di	⊢ (𝜑 → (𝜒 ↔ 𝜃))

Proof of Theorem bitr3di

Step	Hyp	Ref	Expression
1		bitr3di.2	. . 3 ⊢ (𝜓 ↔ 𝜃)
2	1	bicomi 132	. 2 ⊢ (𝜃 ↔ 𝜓)
3		bitr3di.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	bitr2id 193	1 ⊢ (𝜑 → (𝜒 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  xordc  1414  sbal2  2051  eqsnm  3812  fnressn  5798  fressnfv  5799  eluniimadm  5862  iftrueb01  7376  genpassl  7679  genpassu  7680  1idprl  7745  1idpru  7746  axcaucvglemres  8054  negeq0  8368  muleqadd  8783  crap0  9073  addltmul  9316  fzrev  10248  modq0  10518  cjap0  11384  cjne0  11385  caucvgrelemrec  11456  lenegsq  11572  isumss  11868  fsumsplit  11884  sumsplitdc  11909  dvdsabseq  12324  pceu  12784  oddennn  12929  xpsfrnel  13343  metrest  15145  elabgf0  16051