Theorem 3bitr3rd 275

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

Hypotheses

Ref	Expression
3bitr3d.1	⊢ (φ → (ψ ↔ χ))
3bitr3d.2	⊢ (φ → (ψ ↔ θ))
3bitr3d.3	⊢ (φ → (χ ↔ τ))

Assertion

Ref	Expression
3bitr3rd	⊢ (φ → (τ ↔ θ))

Proof of Theorem 3bitr3rd

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

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by: (None)