Theorem syl5rbbr 413

Description: A syllogism inference from two biconditionals.

Hypotheses

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

Assertion

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

Proof of Theorem syl5rbbr

Step	Hyp	Ref	Expression
1		syl5rbbr.1	. 2 ⊢ (φ → (ψ ↔ χ))
2		syl5rbbr.2	. . 3 ⊢ (ψ ↔ θ)
3	2	bicomi 150	. 2 ⊢ (θ ↔ ψ)
4	1, 3	syl5rbb 411	1 ⊢ (φ → (χ ↔ θ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  sbco3 915  sbal2 1005  fniunfv 2860  fressnfv 2898  aceq6b 3565  alephnbtwn2 3675  1idpr 3927  leloet 4284  lerec 4411  nn0subt 4587

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org