Theorem nfeuw 2621

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2622 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2404. (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 1837	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2619	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1577	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

This proof depends on syntax axioms:  ⊤wtru 1571  Ⅎwnf 1816  ∃!weu 2596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2567  df-eu 2597

This theorem is used by:  nfreuw  3399  eusv2nf  5361  reusv2lem3  5366  setrec2  10001  bnj1489  35753  nfalseu  51208