Theorem nfeud2 2582

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) Usage of this theorem is discouraged because it depends on ax-13 2369. Use nfeudw 2583 instead. (New usage is discouraged.)

Hypotheses

Ref	Expression
nfeud2.1	⊢ Ⅎ𝑦𝜑
nfeud2.2	⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfeud2	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud2

Step	Hyp	Ref	Expression
1		df-eu 2561	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
2		nfeud2.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeud2.2	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	2, 3	nfexd2 2443	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfmod2 2550	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)
6	4, 5	nfand 1893	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7	1, 6	nfxfrd 1849	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394  ∀wal 1532  ∃wex 1774  Ⅎwnf 1778  ∃*wmo 2530  ∃!weu 2560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2133  ax-11 2150  ax-12 2170  ax-13 2369

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2532  df-eu 2561

This theorem is referenced by:  nfeud  2584  nfreud  3425