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Theorem exists1 2747
Description: Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see theorem dtru 5248. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.)
Assertion
Ref Expression
exists1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem exists1
StepHypRef Expression
1 equid 2019 . . . 4 𝑥 = 𝑥
21bitru 1547 . . 3 (𝑥 = 𝑥 ↔ ⊤)
32eubii 2669 . 2 (∃!𝑥 𝑥 = 𝑥 ↔ ∃!𝑥⊤)
4 euae 2746 . 2 (∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)
53, 4bitri 278 1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  wtru 1539  ∃!weu 2652
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
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-mo 2622  df-eu 2653
This theorem is referenced by:  exists2  2748
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