Theorem wl-eutf 38485

Description: Closed form of eu6 2600 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 2464	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		nfa1 2188	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3	1, 2	nfan 1932	. 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
4		nfnae 2464	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
5		nfnf1 2191	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
6	5	nfal 2354	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
7	4, 6	nfan 1932	. 2 ⊢ Ⅎ𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑)
8		simpl 488	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → ¬ ∀𝑥 𝑥 = 𝑦)
9		sp 2220	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
10	9	adantl 487	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → Ⅎ𝑦𝜑)
11	3, 7, 8, 10	wl-eudf 38484	1 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  ∃!weu 2594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2213  ax-13 2402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2565  df-eu 2595

This theorem is used by: (None)