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Metamath Proof Explorer

Theorem anbiim

Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024) (Proof shortened by Wolf Lammen, 7-May-2025)

	Ref	Expression
Hypotheses	anbiim.1	$⊢ φ \to (χ \to θ)$
	anbiim.2	$⊢ ψ \to (θ \to χ)$
Assertion	anbiim	$⊢ (φ \land ψ) \to (χ \leftrightarrow θ)$

Proof

Step	Hyp	Ref	Expression
1		anbiim.1	$⊢ φ \to (χ \to θ)$
2		anbiim.2	$⊢ ψ \to (θ \to χ)$
3	1	adantr	$⊢ (φ \land ψ) \to (χ \to θ)$
4	2	adantl	$⊢ (φ \land ψ) \to (θ \to χ)$
5	3 4	impbid	$⊢ (φ \land ψ) \to (χ \leftrightarrow θ)$