Theorem bitr2id 192

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	syl5bb 191	. 2 ⊢ (𝜒 → (𝜑 ↔ 𝜃))
4	3	bicomd 140	1 ⊢ (𝜒 → (𝜃 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bitr3di  194  pm5.17dc  899  dn1dc  955  csbabg  3110  uniiunlem  3236  inimasn  5026  cnvpom  5151  fnresdisj  5306  f1oiso  5802  reldm  6162  mptelixpg  6708  1idprl  7539  1idpru  7540  nndiv  8906  fzn  9985  fz1sbc  10039  metrest  13221