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Metamath Proof Explorer

Theorem nfeuw

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 8-Mar-1995) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

		Ref	Expression
	Hypothesis	nfeuw.1	$⊢ Ⅎ x φ$
	Assertion	nfeuw	$⊢ Ⅎ x \exists! y φ$

Proof

Step	Hyp	Ref	Expression
1		nfeuw.1	$⊢ Ⅎ x φ$
2		nftru	$⊢ Ⅎ y ⊤$
3	1	a1i	$⊢ ⊤ \to Ⅎ x φ$
4	2 3	nfeudw	$⊢ ⊤ \to Ⅎ x \exists! y φ$
5	4	mptru	$⊢ Ⅎ x \exists! y φ$