Theorem 3bitrrd 271

Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3bitrd.1	⊢ (φ → (ψ ↔ χ))
3bitrd.2	⊢ (φ → (χ ↔ θ))
3bitrd.3	⊢ (φ → (θ ↔ τ))

Assertion

Ref	Expression
3bitrrd	⊢ (φ → (τ ↔ ψ))

Proof of Theorem 3bitrrd

Step	Hyp	Ref	Expression
1		3bitrd.3	. 2 ⊢ (φ → (θ ↔ τ))
2		3bitrd.1	. . 3 ⊢ (φ → (ψ ↔ χ))
3		3bitrd.2	. . 3 ⊢ (φ → (χ ↔ θ))
4	2, 3	bitr2d 245	. 2 ⊢ (φ → (θ ↔ ψ))
5	1, 4	bitr3d 246	1 ⊢ (φ → (τ ↔ ψ))

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: (None)