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Metamath Proof Explorer

Theorem nfeud2

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016) (Proof shortened by Wolf Lammen, 4-Oct-2018) (Proof shortened by BJ, 14-Oct-2022) Usage of this theorem is discouraged because it depends on ax-13 . Use nfeudw instead. (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfeud2.1	⊢ Ⅎ 𝑦 𝜑
	nfeud2.2	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
Assertion	nfeud2	⊢ ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfeud2.1	⊢ Ⅎ 𝑦 𝜑
2		nfeud2.2	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
3		df-eu	⊢ ( ∃! 𝑦 𝜓 ↔ ( ∃ 𝑦 𝜓 ∧ ∃* 𝑦 𝜓 ) )
4	1 2	nfexd2	⊢ ( 𝜑 → Ⅎ 𝑥 ∃ 𝑦 𝜓 )
5	1 2	nfmod2	⊢ ( 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 )