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  1280  sbal2  1895  eqsnm  3517  fnressn  5292  fressnfv  5293  eluniimadm  5347  genpassl  6506  genpassu  6507  1idprl  6565  1idpru  6566  negeq0  7041  muleqadd  7411  crap0  7671  addltmul  7918  fzrev  8696  cjap0  9115  cjne0  9116  elabgf0  9231