Theorem wl-exeq 34939

Description: The semantics of ∃𝑥𝑦 = 𝑧. (Contributed by Wolf Lammen, 27-Apr-2018.)

Assertion

Ref	Expression
wl-exeq	⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

Proof of Theorem wl-exeq

Step	Hyp	Ref	Expression
1		nfeqf 2388	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2	1	19.9d 2201	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑥 𝑦 = 𝑧 → 𝑦 = 𝑧))
3	2	impancom 455	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∃𝑥 𝑦 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑧 → 𝑦 = 𝑧))
4	3	orrd 860	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∃𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧))
5	4	expcom 417	. . . . 5 ⊢ (∃𝑥 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
6	5	orrd 860	. . . 4 ⊢ (∃𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
7		3orass 1087	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
8	6, 7	sylibr 237	. . 3 ⊢ (∃𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧))
9		3orrot 1089	. . 3 ⊢ ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧))
10	8, 9	sylibr 237	. 2 ⊢ (∃𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
11		19.8a 2178	. . 3 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
12		ax6e 2390	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑧
13		ax7 2023	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
14	13	com12 32	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑦 = 𝑧))
15	12, 14	eximii 1838	. . . 4 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑦 = 𝑧)
16	15	19.35i 1879	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧)
17		ax6e 2390	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦
18	17, 13	eximii 1838	. . . 4 ⊢ ∃𝑥(𝑥 = 𝑧 → 𝑦 = 𝑧)
19	18	19.35i 1879	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
20	11, 16, 19	3jaoi 1424	. 2 ⊢ ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) → ∃𝑥 𝑦 = 𝑧)
21	10, 20	impbii 212	1 ⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786

This theorem is referenced by:  wl-nfeqfb  34941