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Metamath Proof Explorer

Theorem nfeudw

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu . Version of nfeud with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 15-Feb-2013) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfeudw.1	⊢ Ⅎ 𝑦 𝜑
	nfeudw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfeudw	⊢ ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfeudw.1	⊢ Ⅎ 𝑦 𝜑
2		nfeudw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		df-eu	⊢ ( ∃! 𝑦 𝜓 ↔ ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
4	1 2	nfexd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 𝜓 )
5	1 2	nfmodv	⊢ ( 𝜑 → Ⅎ 𝑥 ∃* 𝑦 𝜓 )
6	4 5	nfand	⊢ ( 𝜑 → Ⅎ 𝑥 ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
7	3 6	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 𝜓 )