Theorem 3bitrrd

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

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

Proof

Step	Hyp	Ref	Expression
1		3bitrd.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		3bitrd.2	$$⊢\phi \to \left(\chi \leftrightarrow \theta \right)$$
3		3bitrd.3	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
4	1 2	bitr2d	$$⊢\phi \to \left(\theta \leftrightarrow \psi \right)$$
5	3 4	bitr3d	$$⊢\phi \to \left(\tau \leftrightarrow \psi \right)$$