Theorem nfeuw 2654

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2655 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 8-Mar-1995.) (Revised by Gino Giotto, 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 1806	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2652	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1545	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  ⊤wtru 1539  Ⅎwnf 1785  ∃!weu 2628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629

This theorem is referenced by:  eusv2nf  5261  reusv2lem3  5266  bnj1489  32438  setrec2  45225