Theorem nfeudw 2583

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2586. Version of nfeud 2584 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 15-Feb-2013.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfeudw.1	⊢ Ⅎ𝑦𝜑
nfeudw.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfeudw	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfeudw

Step	Hyp	Ref	Expression
1		df-eu 2561	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2		nfeudw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeudw.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	2, 3	nfexd 2320	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfmodv 2551	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
6	4, 5	nfand 1898	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7	1, 6	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:   → wi 4   ∧ wa 394  ∃wex 1779  Ⅎwnf 1783  ∃*wmo 2530  ∃!weu 2560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-11 2152  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784  df-mo 2532  df-eu 2561

This theorem is referenced by:  nfeuw  2585  nfreuwOLD  3420