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Metamath Proof Explorer

Theorem nfeud2

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016) (Proof shortened by Wolf Lammen, 4-Oct-2018) (Proof shortened by BJ, 14-Oct-2022) Usage of this theorem is discouraged because it depends on ax-13 . Use nfeudw instead. (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfeud2.1	$⊢ Ⅎ y φ$
	nfeud2.2	$⊢ (φ \land \neg \forall x x = y) \to Ⅎ x ψ$
Assertion	nfeud2	$⊢ φ \to Ⅎ x \exists! y ψ$

Proof

Step	Hyp	Ref	Expression
1		nfeud2.1	$⊢ Ⅎ y φ$
2		nfeud2.2	$⊢ (φ \land \neg \forall x x = y) \to Ⅎ x ψ$
3		df-eu	$⊢ \exists! y ψ \leftrightarrow (\exists y ψ \land \exists^{*} y ψ)$
4	1 2	nfexd2	$⊢ φ \to Ⅎ x \exists y ψ$
5	1 2	nfmod2	$⊢ φ \to Ⅎ x \exists^{*} y ψ$
6	4 5	nfand	$⊢ φ \to Ⅎ x (\exists y ψ \land \exists^{*} y ψ)$
7	3 6	nfxfrd	$⊢ φ \to Ⅎ x \exists! y ψ$