Theorem bibi12d

Description: Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993)

	Ref	Expression
Hypotheses	imbi12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
	imbi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
Assertion	bibi12d	$$⊢\phi \to \left(\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \tau \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imbi12d.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2		imbi12d.2	$$⊢\phi \to \left(\theta \leftrightarrow \tau \right)$$
3	1	bibi1d	$$⊢\phi \to \left(\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \theta \right)\right)$$
4	2	bibi2d	$$⊢\phi \to \left(\left(\chi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \tau \right)\right)$$
5	3 4	bitrd	$$⊢\phi \to \left(\left(\psi \leftrightarrow \theta \right)\leftrightarrow \left(\chi \leftrightarrow \tau \right)\right)$$