Theorem 3bitr2rd 273

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

Hypotheses

Ref	Expression
3bitr2d.1	⊢ (φ → (ψ ↔ χ))
3bitr2d.2	⊢ (φ → (θ ↔ χ))
3bitr2d.3	⊢ (φ → (θ ↔ τ))

Assertion

Ref	Expression
3bitr2rd	⊢ (φ → (τ ↔ ψ))

Proof of Theorem 3bitr2rd

Step	Hyp	Ref	Expression
1		3bitr2d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		3bitr2d.2	. . 3 ⊢ (φ → (θ ↔ χ))
3	1, 2	bitr4d 247	. 2 ⊢ (φ → (ψ ↔ θ))
4		3bitr2d.3	. 2 ⊢ (φ → (θ ↔ τ))
5	3, 4	bitr2d 245	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:  eqtfinrelk  4487