Theorem 3bitr2rd

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

	Ref	Expression
Hypotheses	3bitr2d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	3bitr2d.2	$$⊢\phi \to \left(\theta \leftrightarrow \chi \right)$$
	3bitr2d.3	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
Assertion	3bitr2rd	$$⊢\phi \to \left(\tau \leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		3bitr2d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		3bitr2d.2	$$⊢\phi \to \left(\theta \leftrightarrow \chi \right)$$
3		3bitr2d.3	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
4	1 2	bitr4d	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
5	4 3	bitr2d	$$⊢\phi \to \left(\tau \leftrightarrow \psi \right)$$