Theorem 3bitr4d

Description: Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995)

	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	3bitr4d	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$

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	1 3	bitr4d	$$⊢\phi \to \left(\psi \leftrightarrow \tau \right)$$
5	2 4	bitrd	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$