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Theorem wl-euae 34626
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 2652 . 2 (∃!𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
2 extru 1973 . . 3 𝑥
32biantrur 531 . 2 (∃*𝑥⊤ ↔ (∃𝑥⊤ ∧ ∃*𝑥⊤))
4 wl-moae 34625 . 2 (∃*𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
51, 3, 43bitr2i 300 1 (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wal 1528  wtru 1531  wex 1773  ∃*wmo 2618  ∃!weu 2651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-mo 2620  df-eu 2652
This theorem is referenced by: (None)
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