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Metamath Proof Explorer

Theorem nfeu

Description: Bound-variable hypothesis builder for the unique existential quantifier. Note that x and y need not be disjoint. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfeuw when possible. (Contributed by NM, 8-Mar-1995) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	nfeu.1	⊢ Ⅎ 𝑥 𝜑
	Assertion	nfeu	⊢ Ⅎ 𝑥 ∃! 𝑦 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfeu.1	⊢ Ⅎ 𝑥 𝜑
2		nftru	⊢ Ⅎ 𝑦 ⊤
3	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
4	2 3	nfeud	⊢ ( ⊤ → Ⅎ 𝑥 ∃! 𝑦 𝜑 )
5	4	mptru	⊢ Ⅎ 𝑥 ∃! 𝑦 𝜑