Theorem nfeud

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfeudw when possible. (Contributed by NM, 15-Feb-2013) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfeud.1	⊢ Ⅎ 𝑦 𝜑
2		nfeud.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3	2	adantr	⊢ ( ( 𝜑 ∧ ¬ ∀ 𝑥 𝑥 = 𝑦 ) → Ⅎ 𝑥 𝜓 )
4	1 3	nfeud2	⊢ ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 𝜓 )