Theorem 3bitr3d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996)

	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	3bitr3d	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$

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	2 1	bitr3d	$$⊢\phi \to \left(\theta \leftrightarrow \chi \right)$$
5	4 3	bitrd	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$