Theorem mpbi2and

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011) (Proof shortened by Wolf Lammen, 24-Nov-2012)

	Ref	Expression
Hypotheses	mpbi2and.1	$$⊢\phi \to \psi$$
	mpbi2and.2	$$⊢\phi \to \chi$$
	mpbi2and.3	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)$$
Assertion	mpbi2and	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		mpbi2and.1	$$⊢\phi \to \psi$$
2		mpbi2and.2	$$⊢\phi \to \chi$$
3		mpbi2and.3	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \theta \right)$$
4	1 2	jca	$$⊢\phi \to \left(\psi \wedge \chi \right)$$
5	4 3	mpbid	$$⊢\phi \to \theta$$