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Theorem euequ1 2043
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 1631 . 2 𝑥 𝑥 = 𝑦
2 equtr2 1644 . . 3 ((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
32gen2 1384 . 2 𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)
4 equequ1 1645 . . 3 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
54eu4 2010 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝑧((𝑥 = 𝑦𝑧 = 𝑦) → 𝑥 = 𝑧)))
61, 3, 5mpbir2an 888 1 ∃!𝑥 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  copsexg  4062  oprabid  5663
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