Theorem impbid

Description: Deduce an equivalence from two implications. Deduction associated with impbi and impbii . (Contributed by NM, 24-Jan-1993) Revised to prove it from impbid21d . (Revised by Wolf Lammen, 3-Nov-2012)

	Ref	Expression
Hypotheses	impbid.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
	impbid.2	$$⊢\phi \to \left(\chi \to \psi \right)$$
Assertion	impbid	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		impbid.1	$$⊢\phi \to \left(\psi \to \chi \right)$$
2		impbid.2	$$⊢\phi \to \left(\chi \to \psi \right)$$
3	1 2	impbid21d	$$⊢\phi \to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$
4	3	pm2.43i	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$