Theorem syl5rbb 249

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 248	. 2 ⊢ (χ → (φ ↔ θ))
4	3	bicomd 192	1 ⊢ (χ → (θ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  syl5rbbr  251  csbabg  3197  uniiunlem  3353  opkelimagekg  4271  setswith  4321  fnresdisj  5193  f1oiso  5499