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Theorem eueq 2689
 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 2041 . . . 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 2539 . 2 (A V ↔ x x = A)
5 eqeq1 2028 . . 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 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃!weu 1882  Vcvv 2535 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  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537 This theorem is referenced by:  eueq1  2690  moeq  2693  mosubt  2695  reuhypd  4153  mptfng  4950
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