Theorem wl-exeq 32500

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 2289	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2	1	19.9d 2058	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑥 𝑦 = 𝑧 → 𝑦 = 𝑧))
3	2	impancom 455	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∃𝑥 𝑦 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑧 → 𝑦 = 𝑧))
4	3	orrd 392	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∃𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧))
5	4	expcom 450	. . . 4 ⊢ (∃𝑥 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
6	5	orrd 392	. . 3 ⊢ (∃𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
7		3orrot 1037	. . . 4 ⊢ ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧))
8		3orass 1034	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
9	7, 8	bitri 263	. . 3 ⊢ ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ 𝑦 = 𝑧)))
10	6, 9	sylibr 223	. 2 ⊢ (∃𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
11		19.8a 2039	. . 3 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
12		ax6e 2238	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑧
13		ax7 1930	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
14	13	com12 32	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑦 = 𝑧))
15	12, 14	eximii 1754	. . . 4 ⊢ ∃𝑥(𝑥 = 𝑦 → 𝑦 = 𝑧)
16	15	19.35i 1795	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧)
17		ax6e 2238	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦
18	17, 13	eximii 1754	. . . 4 ⊢ ∃𝑥(𝑥 = 𝑧 → 𝑦 = 𝑧)
19	18	19.35i 1795	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
20	11, 16, 19	3jaoi 1383	. 2 ⊢ ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) → ∃𝑥 𝑦 = 𝑧)
21	10, 20	impbii 198	1 ⊢ (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  wl-nfeqfb  32502