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Theorem eueq 2706
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 2056 . . . 4 ((x = A y = A) → x = y)
21gen2 1336 . . 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 2555 . 2 (A V ↔ x x = A)
5 eqeq1 2043 . . 3 (x = y → (x = Ay = A))
65eu4 1959 . 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 1240   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  Vcvv 2551
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  eueq1  2707  moeq  2710  mosubt  2712  reuhypd  4169  mptfng  4967
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