Theorem wl-eutf 35728

Description: Closed form of eu6 2574 with a distinctor avoiding distinct variable conditions. (Contributed by Wolf Lammen, 23-Sep-2020.)

Assertion

Ref	Expression
wl-eutf	⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Proof of Theorem wl-eutf

Step	Hyp	Ref	Expression
1		nfnae 2434	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		nfa1 2148	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3	1, 2	nfan 1902	. 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
4		nfnae 2434	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5		nfnf1 2151	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
6	5	nfal 2317	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
7	4, 6	nfan 1902	. 2 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
8		simpl 483	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9		sp 2176	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
10	9	adantl 482	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → Ⅎ𝑦𝜑)
11	3, 7, 8, 10	wl-eudf 35727	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)