Theorem mpbiran3d

Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		mpbiran3d.1	$$⊢\phi \to \left(\psi \leftrightarrow \left(\chi \wedge \theta \right)\right)$$
2		mpbiran3d.2	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
3	1	simprbda	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
4	3	ex	$$⊢\phi \to \left(\psi \to \chi \right)$$
5	2	ex	$$⊢\phi \to \left(\chi \to \theta \right)$$
6	5	ancld	$$⊢\phi \to \left(\chi \to \left(\chi \wedge \theta \right)\right)$$
7	6 1	sylibrd	$$⊢\phi \to \left(\chi \to \psi \right)$$
8	4 7	impbid	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$