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Theorem euequ1 2092
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 1674 . 2 𝑥 𝑥 = 𝑦
2 equtr2 1687 . . 3 ((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
32gen2 1426 . 2 𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
4 equequ1 1688 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
54eu4 2059 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)))
61, 3, 5mpbir2an 926 1 ∃!𝑥 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329  wex 1468  ∃!weu 1997
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001
This theorem is referenced by:  copsexg  4161  oprabid  5796
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