Theorem nfreuwOLD

Description: Obsolete version of nfreuw as of 21-Nov-2024. (Contributed by NM, 30-Oct-2010) (Revised by GG, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfreuwOLD.1	⊢ Ⅎ 𝑥 𝐴
	nfreuwOLD.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfreuwOLD	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfreuwOLD.1	⊢ Ⅎ 𝑥 𝐴
2		nfreuwOLD.2	⊢ Ⅎ 𝑥 𝜑
3		df-reu	⊢ ( ∃! 𝑦 ∈ 𝐴 𝜑 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
4		nftru	⊢ Ⅎ 𝑦 ⊤
5		nfcvd	⊢ ( ⊤ → Ⅎ 𝑥 𝑦 )
6	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
7	5 6	nfeld	⊢ ( ⊤ → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
8	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
9	7 8	nfand	⊢ ( ⊤ → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
10	4 9	nfeudw	⊢ ( ⊤ → Ⅎ 𝑥 ∃! 𝑦 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
11	3 10	nfxfrd	⊢ ( ⊤ → Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑 )
12	11	mptru	⊢ Ⅎ 𝑥 ∃! 𝑦 ∈ 𝐴 𝜑