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Metamath Proof Explorer

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	$⊢ φ \to (ψ \leftrightarrow (χ \land θ))$
	mpbiran4d.2	$⊢ (φ \land θ) \to χ$
Assertion	mpbiran4d	$⊢ φ \to (ψ \leftrightarrow θ)$

Proof

Step	Hyp	Ref	Expression
1		mpbiran3d.1	$⊢ φ \to (ψ \leftrightarrow (χ \land θ))$
2		mpbiran4d.2	$⊢ (φ \land θ) \to χ$
3	1	biancomd	$⊢ φ \to (ψ \leftrightarrow (θ \land χ))$
4	3 2	mpbiran3d	$⊢ φ \to (ψ \leftrightarrow θ)$