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Theorem euequ1 1977
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 1568 . 2 x x = y
2 equtr2 1579 . . 3 ((x = y z = y) → x = z)
32gen2 1319 . 2 xz((x = y z = y) → x = z)
4 equequ1 1580 . . 3 (x = z → (x = yz = y))
54eu4 1944 . 2 (∃!x x = y ↔ (x x = y xz((x = y z = y) → x = z)))
61, 3, 5mpbir2an 837 1 ∃!x x = y
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226  wex 1362  ∃!weu 1882
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886
This theorem is referenced by:  copsexg  3955  oprabid  5461
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