Theorem nfeud2 2482

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfeud2	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2474	. 2 ⊢ (∃!𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
2		nfv 1843	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfeud2.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfeud2.2	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
5		nfeqf1 2299	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
6	5	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧)
7	4, 6	nfbid 1832	. . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 ↔ 𝑦 = 𝑧))
8	3, 7	nfald2 2331	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
9	2, 8	nfexd 2167	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 ↔ 𝑦 = 𝑧))
10	1, 9	nfxfrd 1780	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708  ∃!weu 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474

This theorem is referenced by:  nfmod2  2483  nfeud  2484  nfreud  3112