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Metamath Proof Explorer

Theorem nfeu1

Description: Bound-variable hypothesis builder for uniqueness. See also nfeu1ALT . (Contributed by NM, 9-Jul-1994) (Revised by Mario Carneiro, 7-Oct-2016)

		Ref	Expression
	Assertion	nfeu1	$⊢ Ⅎ x \exists! x φ$

Proof

Step	Hyp	Ref	Expression
1		eu6	$⊢ \exists! x φ \leftrightarrow \exists y \forall x (φ \leftrightarrow x = y)$
2		nfa1	$⊢ Ⅎ x \forall x (φ \leftrightarrow x = y)$
3	2	nfex	$⊢ Ⅎ x \exists y \forall x (φ \leftrightarrow x = y)$
4	1 3	nfxfr	$⊢ Ⅎ x \exists! x φ$