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Theorem wl-euae 33840
 Description: Two ways to express "exactly one thing exists" . (Contributed by Wolf Lammen, 5-Mar-2023.)
Assertion
Ref Expression
wl-euae (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem wl-euae
StepHypRef Expression
1 df-eu 2640 . 2 (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
2 extru 2079 . . 3 𝑥
32biantrur 526 . 2 (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
4 wl-moae 33839 . 2 (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
51, 3, 43bitr2i 291 1 (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386  ∀wal 1654  ⊤wtru 1657  ∃wex 1878  ∃*wmo 2603  ∃!weu 2639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112 This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1660  df-ex 1879  df-mo 2605  df-eu 2640 This theorem is referenced by: (None)
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