Theorem nfeuw 2579

Description: Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2580 with a disjoint variable condition, which does not require ax-13 2363. (Contributed by NM, 8-Mar-1995.) Avoid ax-13 2363. (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 1798	. . 3 ⊢ Ⅎ𝑦⊤
2		nfeuw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfeudw 2577	. 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦𝜑)
5	4	mptru 1540	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  ⊤wtru 1534  Ⅎwnf 1777  ∃!weu 2554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-mo 2526  df-eu 2555

This theorem is referenced by:  nfreuw  3402  eusv2nf  5384  reusv2lem3  5389  bnj1489  34585  setrec2  47987