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Theorem exists1 2012
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. (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 1919 . 2 (∃!𝑥 𝑥 = 𝑥 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
2 equid 1605 . . . . . 6 𝑥 = 𝑥
32tbt 240 . . . . 5 (𝑥 = 𝑦 ↔ (𝑥 = 𝑦𝑥 = 𝑥))
4 bicom 132 . . . . 5 ((𝑥 = 𝑦𝑥 = 𝑥) ↔ (𝑥 = 𝑥𝑥 = 𝑦))
53, 4bitri 177 . . . 4 (𝑥 = 𝑦 ↔ (𝑥 = 𝑥𝑥 = 𝑦))
65albii 1375 . . 3 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥(𝑥 = 𝑥𝑥 = 𝑦))
76exbii 1512 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∃𝑦𝑥(𝑥 = 𝑥𝑥 = 𝑦))
8 hbae 1622 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
9819.9h 1550 . 2 (∃𝑦𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
101, 7, 93bitr2i 201 1 (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257  wex 1397  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-eu 1919
This theorem is referenced by:  exists2  2013
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