Theorem syl5rbb 191

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

Hypotheses

Ref	Expression
syl5rbb.1	⊢ (𝜑 ↔ 𝜓)
syl5rbb.2	⊢ (𝜒 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
syl5rbb	⊢ (𝜒 → (𝜃 ↔ 𝜑))

Proof of Theorem syl5rbb

Step	Hyp	Ref	Expression
1		syl5rbb.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2		syl5rbb.2	. . 3 ⊢ (𝜒 → (𝜓 ↔ 𝜃))
3	1, 2	syl5bb 190	. 2 ⊢ (𝜒 → (𝜑 ↔ 𝜃))
4	3	bicomd 139	1 ⊢ (𝜒 → (𝜃 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  syl5rbbr  193  pm5.17dc  848  dn1dc  906  csbabg  2989  uniiunlem  3109  inimasn  4849  cnvpom  4973  fnresdisj  5124  f1oiso  5605  reldm  5956  1idprl  7147  1idpru  7148  nndiv  8461  fzn  9454  fz1sbc  9506  bj-indeq  11779