Theorem syl5rbbr 184

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 123	. 2 ⊢ (φ ↔ ψ)
3		syl5rbbr.2	. 2 ⊢ (χ → (ψ ↔ θ))
4	2, 3	syl5rbb 182	1 ⊢ (χ → (θ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  xordc  1266  sbal2  1880  eqsnm  3500  fnressn  5274  fressnfv  5275  eluniimadm  5329  genpassl  6379  genpassu  6380  1idprl  6429  1idpru  6430  negeq0  6851  elabgf0  7023