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  908  dn1dc  965  csbabg  3166  uniiunlem  3293  inimasn  5122  cnvpom  5247  fnresdisj  5409  f1oiso  5923  reldm  6302  mptelixpg  6851  1idprl  7745  1idpru  7746  nndiv  9119  fzn  10206  fz1sbc  10260  grpid  13538  znleval  14582  metrest  15145