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  905  dn1dc  962  csbabg  3146  uniiunlem  3273  inimasn  5088  cnvpom  5213  fnresdisj  5371  f1oiso  5876  reldm  6253  mptelixpg  6802  1idprl  7674  1idpru  7675  nndiv  9048  fzn  10134  fz1sbc  10188  grpid  13241  znleval  14285  metrest  14826