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Theorem euequ1 1995
 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 1586 . 2 𝑥 𝑥 = 𝑦
2 equtr2 1597 . . 3 ((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
32gen2 1339 . 2 𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
4 equequ1 1598 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
54eu4 1962 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)))
61, 3, 5mpbir2an 849 1 ∃!𝑥 𝑥 = 𝑦
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381  ∃!weu 1900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904 This theorem is referenced by:  copsexg  3981  oprabid  5537
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