Theorem mpbir3and

Description: Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014) (Revised by Mario Carneiro, 9-Jan-2015)

	Ref	Expression
Hypotheses	mpbir3and.1	$$⊢\phi \to \chi$$
	mpbir3and.2	$$⊢\phi \to \theta$$
	mpbir3and.3	$$⊢\phi \to \tau$$
	mpbir3and.4	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \wedge \tau \right)\right)$$
Assertion	mpbir3and	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		mpbir3and.1	$$⊢\phi \to \chi$$
2		mpbir3and.2	$$⊢\phi \to \theta$$
3		mpbir3and.3	$$⊢\phi \to \tau$$
4		mpbir3and.4	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \wedge \tau \right)\right)$$
5	1 2 3	3jca	$$⊢\phi \to \left(\chi \wedge \theta \wedge \tau \right)$$
6	5 4	mpbird	$$⊢\phi \to \psi$$