Metamath Proof Explorer


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 𝑦 𝜑
nfeud.2 ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion nfeud ( 𝜑 → Ⅎ 𝑥 ∃! 𝑦 𝜓 )

Proof

Step Hyp Ref Expression
1 nfeud.1 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 )