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Theorem eueq 2688
Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eueq (A V ↔ ∃!x x = A)
Distinct variable group:   x,A

Proof of Theorem eueq
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2042 . . . 4 ((x = A y = A) → x = y)
21gen2 1319 . . 3 xy((x = A y = A) → x = y)
32biantru 286 . 2 (x x = A ↔ (x x = A xy((x = A y = A) → x = y)))
4 isset 2538 . 2 (A V ↔ x x = A)
5 eqeq1 2029 . . 3 (x = y → (x = Ay = A))
65eu4 1944 . 2 (∃!x x = A ↔ (x x = A xy((x = A y = A) → x = y)))
73, 4, 63bitr4i 201 1 (A V ↔ ∃!x x = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1315  wex 1362   = wceq 1374   wcel 1376  ∃!weu 1882  Vcvv 2534
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 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1629  df-eu 1885  df-mo 1886  df-clab 2010  df-cleq 2016  df-clel 2019  df-v 2536
This theorem is referenced by:  eueq1  2689  moeq  2692  mosubt  2694  reuhypd  4129  mptfng  4927
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