Theorem mpbir2and

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	mpbir2and.1	$$⊢\phi \to \chi$$
	mpbir2and.2	$$⊢\phi \to \theta$$
	mpbir2and.3	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
Assertion	mpbir2and	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		mpbir2and.1	$$⊢\phi \to \chi$$
2		mpbir2and.2	$$⊢\phi \to \theta$$
3		mpbir2and.3	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
4	1 2	jca	$$⊢\phi \to \left(\chi \wedge \theta \right)$$
5	4 3	mpbird	$$⊢\phi \to \psi$$