Theorem nfeuw 2628

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2629 with a disjoint variable condition, which does not require ax-13 2411. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2411. (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 1832	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2626	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1575	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  ⊤wtru 1569  Ⅎwnf 1811  ∃!weu 2603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-10 2183  ax-11 2199  ax-12 2220

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-mo 2574  df-eu 2604

This theorem is referenced by:  nfreuw  3406  eusv2nf  5370  reusv2lem3  5375  bnj1489  35414  setrec2  50422