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Theorem exists1 2560
Description: Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4827. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
exists1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦

Proof of Theorem exists1
StepHypRef Expression
1 df-eu 2473 . 2 (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
2 equid 1936 . . . . . 6 𝑥 = 𝑥
32tbt 359 . . . . 5 (𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥 = 𝑥))
4 bicom 212 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝑥) ↔ (𝑥 = 𝑥𝑥 = 𝑦))
53, 4bitri 264 . . . 4 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
65albii 1744 . . 3 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥𝑥 = 𝑦))
76exbii 1771 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
8 nfae 2315 . . 3 𝑦𝑥 𝑥 = 𝑦
9819.9 2070 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
101, 7, 93bitr2i 288 1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1478  wex 1701  ∃!weu 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2473
This theorem is referenced by:  exists2  2561
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