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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	⊢ Ⅎ 𝑥 𝐴
	nfrmo.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfreu	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfrmo.1	⊢ Ⅎ 𝑥 𝐴
2		nfrmo.2	⊢ Ⅎ 𝑥 𝜑
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
6	3 4 5	nfreud	⊢ ( ⊤ → Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑 )
7	6	mptru	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑