Theorem mpbiran4d

Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024)

	Ref	Expression
Hypotheses	mpbiran3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
	mpbiran4d.2	$$⊢\left(\phi \wedge \theta \right)\to \chi$$
Assertion	mpbiran4d	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		mpbiran3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
2		mpbiran4d.2	$$⊢\left(\phi \wedge \theta \right)\to \chi$$
3	1	biancomd	$$⊢\phi \to \left(\psi \leftrightarrow \left(\theta \wedge \chi \right)\right)$$
4	3 2	mpbiran3d	$$⊢\phi \to \left(\psi \leftrightarrow \theta \right)$$