Theorem wl-eutf 38083

Description: Closed form of eu6 2604 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 2468	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		nfa1 2188	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3	1, 2	nfan 1922	. 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
4		nfnae 2468	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5		nfnf1 2191	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
6	5	nfal 2358	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
7	4, 6	nfan 1922	. 2 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
8		simpl 487	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9		sp 2221	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
10	9	adantl 486	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → Ⅎ𝑦𝜑)
11	3, 7, 8, 10	wl-eudf 38082	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  ∃wex 1802  Ⅎwnf 1806  ∃!weu 2598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-mo 2569  df-eu 2599

This theorem is referenced by: (None)