Theorem 3bitrd

Description: Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999)

	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	3bitrd	$$⊢\phi \to \left(\psi \leftrightarrow \tau \right)$$

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	bitrd	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$
5	4 3	bitrd	$$⊢\phi \to \left(\psi \leftrightarrow \tau \right)$$