Theorem syl5rbbr 251

Description: A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Hypotheses

Ref	Expression
syl5rbbr.1	⊢ (ψ ↔ φ)
syl5rbbr.2	⊢ (χ → (ψ ↔ θ))

Assertion

Ref	Expression
syl5rbbr	⊢ (χ → (θ ↔ φ))

Proof of Theorem syl5rbbr

Step	Hyp	Ref	Expression
1		syl5rbbr.1	. . 3 ⊢ (ψ ↔ φ)
2	1	bicomi 193	. 2 ⊢ (φ ↔ ψ)
3		syl5rbbr.2	. 2 ⊢ (χ → (ψ ↔ θ))
4	2, 3	syl5rbb 249	1 ⊢ (χ → (θ ↔ φ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  sbco3  2088  sbal2  2134  dmfco  5382  fressnfv  5440  eluniima  5470  txpcofun  5804  brfullfung  5866