Theorem nfeuw 2618

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2619 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2401. (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 1836	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2616	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1576	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

This proof depends on syntax axioms:  ⊤wtru 1570  Ⅎwnf 1815  ∃!weu 2593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-mo 2564  df-eu 2594

This theorem is used by:  nfreuw  3395  eusv2nf  5356  reusv2lem3  5361  setrec2  9977  bnj1489  35686  nfalseu  50930