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Metamath Proof Explorer

Theorem reu6dv

Description: A condition which implies existential uniqueness. (Contributed by Thierry Arnoux, 13-Oct-2025)

	Ref	Expression
Hypotheses	reu6d.1	$⊢ φ \to B \in A$
	reu6d.2	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow x = B)$
Assertion	reu6dv	$⊢ φ \to \exists! x \in A ψ$

Step	Hyp	Ref	Expression
1		reu6d.1	$⊢ φ \to B \in A$
2		reu6d.2	$⊢ (φ \land x \in A) \to (ψ \leftrightarrow x = B)$
3	2	ralrimiva	$⊢ φ \to \forall x \in A (ψ \leftrightarrow x = B)$
4		reu6i	$⊢ (B \in A \land \forall x \in A (ψ \leftrightarrow x = B)) \to \exists! x \in A ψ$
5	1 3 4	syl2anc	$⊢ φ \to \exists! x \in A ψ$