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Theorem euequ1 2114
Description: Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
Assertion
Ref Expression
euequ1 ∃!𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem euequ1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 a9e 1689 . 2 𝑥 𝑥 = 𝑦
2 equtr2 1704 . . 3 ((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
32gen2 1443 . 2 𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
4 equequ1 1705 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
54eu4 2081 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)))
61, 3, 5mpbir2an 937 1 ∃!𝑥 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1346  wex 1485  ∃!weu 2019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023
This theorem is referenced by:  copsexg  4227  oprabid  5883
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