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Theorem wl-exeq 33451
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
StepHypRef Expression
1 nfeqf 2337 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦 = 𝑧)
2119.9d 2108 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑥 𝑦 = 𝑧𝑦 = 𝑧))
32impancom 455 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∃𝑥 𝑦 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑧𝑦 = 𝑧))
43orrd 392 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∃𝑥 𝑦 = 𝑧) → (∀𝑥 𝑥 = 𝑧𝑦 = 𝑧))
54expcom 450 . . . 4 (∃𝑥 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧𝑦 = 𝑧)))
65orrd 392 . . 3 (∃𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧𝑦 = 𝑧)))
7 3orrot 1061 . . . 4 ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧𝑦 = 𝑧))
8 3orass 1057 . . . 4 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧𝑦 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧𝑦 = 𝑧)))
97, 8bitri 264 . . 3 ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ↔ (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧𝑦 = 𝑧)))
106, 9sylibr 224 . 2 (∃𝑥 𝑦 = 𝑧 → (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
11 19.8a 2090 . . 3 (𝑦 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
12 ax6e 2286 . . . . 5 𝑥 𝑥 = 𝑧
13 ax7 1989 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
1413com12 32 . . . . 5 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑦 = 𝑧))
1512, 14eximii 1804 . . . 4 𝑥(𝑥 = 𝑦𝑦 = 𝑧)
161519.35i 1846 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧)
17 ax6e 2286 . . . . 5 𝑥 𝑥 = 𝑦
1817, 13eximii 1804 . . . 4 𝑥(𝑥 = 𝑧𝑦 = 𝑧)
191819.35i 1846 . . 3 (∀𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
2011, 16, 193jaoi 1431 . 2 ((𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) → ∃𝑥 𝑦 = 𝑧)
2110, 20impbii 199 1 (∃𝑥 𝑦 = 𝑧 ↔ (𝑦 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3o 1053  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  wl-nfeqfb  33453
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