Theorem wl-eutf 33355

Description: Closed form of df-eu 2474 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 2318	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		nfa1 2028	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3	1, 2	nfan 1828	. 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
4		nfnae 2318	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5		nfnf1 2031	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
6	5	nfal 2153	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
7	4, 6	nfan 1828	. 2 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
8		simpl 473	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9		sp 2053	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
10	9	adantl 482	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → Ⅎ𝑦𝜑)
11	3, 7, 8, 10	wl-eudf 33354	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

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: (None)