Theorem bitr2id 193

Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
bitr2id.1	⊢ (𝜑 ↔ 𝜓)
bitr2id.2	⊢ (𝜒 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
bitr2id	⊢ (𝜒 → (𝜃 ↔ 𝜑))

Proof of Theorem bitr2id

Step	Hyp	Ref	Expression
1		bitr2id.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		bitr2id.2	. . 3 ⊢ (𝜒 → (𝜓 ↔ 𝜃))
3	1, 2	bitrid 192	. 2 ⊢ (𝜒 → (𝜑 ↔ 𝜃))
4	3	bicomd 141	1 ⊢ (𝜒 → (𝜃 ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  bitr3di  195  pm5.17dc  916  dn1dc  973  csbabg  3209  uniiunlem  3338  inimasn  5205  cnvpom  5330  fnresdisj  5493  f1oiso  6032  reldm  6420  mptelixpg  7016  1idprl  7957  1idpru  7958  nndiv  9346  fzn  10448  fz1sbc  10505  grpid  13846  znleval  14990  metrest  15609  loopclwwlkn1b  16672  clwwlknun  16694