Theorem nfmodv

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod for a version without disjoint variable conditions but requiring ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016) (Revised by BJ, 28-Jan-2023)

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

Proof

Step	Hyp	Ref	Expression
1		nfmodv.1	⊢ Ⅎ 𝑦 𝜑
2		nfmodv.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		df-mo	⊢ ( ∃* 𝑦 𝜓 ↔ ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) )
4		nfv	⊢ Ⅎ 𝑧 𝜑
5		nfvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 = 𝑧 )
6	2 5	nfimd	⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝜓 → 𝑦 = 𝑧 ) )
7	1 6	nfald	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) )
8	4 7	nfexd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑧 ∀ 𝑦 ( 𝜓 → 𝑦 = 𝑧 ) )
9	3 8	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∃* 𝑦 𝜓 )