Theorem nfeuw 2622

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2623 with a disjoint variable condition, which does not require ax-13 2405. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2405. (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 1826	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2620	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1569	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  ⊤wtru 1563  Ⅎwnf 1805  ∃!weu 2597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-11 2193  ax-12 2214

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-mo 2568  df-eu 2598

This theorem is referenced by:  nfreuw  3399  eusv2nf  5354  reusv2lem3  5359  bnj1489  35353  setrec2  50321