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))

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:  bitr3di  195  pm5.17dc  909  dn1dc  966  csbabg  3186  uniiunlem  3313  inimasn  5146  cnvpom  5271  fnresdisj  5433  f1oiso  5956  reldm  6338  mptelixpg  6889  1idprl  7785  1idpru  7786  nndiv  9159  fzn  10246  fz1sbc  10300  grpid  13580  znleval  14625  metrest  15188