Theorem wl-eutf 37527

Description: Closed form of eu6 2577 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 2442	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		nfa1 2152	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3	1, 2	nfan 1898	. 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
4		nfnae 2442	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5		nfnf1 2155	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
6	5	nfal 2327	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
7	4, 6	nfan 1898	. 2 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
8		simpl 482	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9		sp 2184	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
10	9	adantl 481	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → Ⅎ𝑦𝜑)
11	3, 7, 8, 10	wl-eudf 37526	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-eu 2572

This theorem is referenced by: (None)