Theorem nfeuw 2620

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2621 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2403. (Revised by GG, 10-Jan-2024.)

Hypothesis

Ref	Expression
nfeuw.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfeuw	⊢ Ⅎ𝑥∃!𝑦𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem nfeuw

Step	Hyp	Ref	Expression
1		nftru 1833	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2618	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1576	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

This proof depends on syntax axioms:  ⊤wtru 1570  Ⅎwnf 1812  ∃!weu 2595

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-11 2191  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-mo 2566  df-eu 2596

This theorem is used by:  nfreuw  3398  eusv2nf  5365  reusv2lem3  5370  bnj1489  35453  setrec2  50501  nfalseu  50640