Theorem 3bitr4rd

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

	Ref	Expression
Hypotheses	3bitr4d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	3bitr4d.2	$$⊢\phi \to \left(\theta \leftrightarrow \psi \right)$$
	3bitr4d.3	$$⊢\phi \to \left(\tau \leftrightarrow \chi \right)$$
Assertion	3bitr4rd	$$⊢\phi \to \left(\tau \leftrightarrow \theta \right)$$

Proof

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