Theorem impbida

Description: Deduce an equivalence from two implications. Variant of impbid . (Contributed by NM, 17-Feb-2007)

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

Proof

Step	Hyp	Ref	Expression
1		impbida.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		impbida.2	$$⊢\left(\phi \wedge \chi \right)\to \psi$$
3	1	ex	$$⊢\phi \to \left(\psi \to \chi \right)$$
4	2	ex	$$⊢\phi \to \left(\chi \to \psi \right)$$
5	3 4	impbid	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$