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Metamath Proof Explorer

Theorem nfreu

Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfreuw when possible. (Contributed by NM, 30-Oct-2010) (Revised by Mario Carneiro, 8-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfrmo.1	$⊢ \underline{Ⅎ} x A$
	nfrmo.2	$⊢ Ⅎ x φ$
Assertion	nfreu	$⊢ Ⅎ x \exists! y \in A φ$

Proof

Step	Hyp	Ref	Expression
1		nfrmo.1	$⊢ \underline{Ⅎ} x A$
2		nfrmo.2	$⊢ Ⅎ x φ$
3		nftru	$⊢ Ⅎ y ⊤$
4	1	a1i	$⊢ ⊤ \to \underline{Ⅎ} x A$
5	2	a1i	$⊢ ⊤ \to Ⅎ x φ$
6	3 4 5	nfreud	$⊢ ⊤ \to Ⅎ x \exists! y \in A φ$
7	6	mptru	$⊢ Ⅎ x \exists! y \in A φ$