Theorem 3bitr3rd

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

	Ref	Expression
Hypotheses	3bitr3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	3bitr3d.2	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
	3bitr3d.3	$$⊢\phi \to \left(\chi \leftrightarrow \tau \right)$$
Assertion	3bitr3rd	$$⊢\phi \to \left(\tau \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		3bitr3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		3bitr3d.2	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
3		3bitr3d.3	$$⊢\phi \to \left(\chi \leftrightarrow \tau \right)$$
4	1 2	bitr3d	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$
5	3 4	bitr3d	$$⊢\phi \to \left(\tau \leftrightarrow \theta \right)$$