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Theorem eusv1 4150
Description: Two ways to express single-valuedness of a class expression A(x). (Contributed by NM, 14-Oct-2010.)
Assertion
Ref Expression
eusv1 (∃!yx y = Ayx y = A)
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusv1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sp 1398 . . . 4 (x y = Ay = A)
2 sp 1398 . . . 4 (x z = Az = A)
3 eqtr3 2056 . . . 4 ((y = A z = A) → y = z)
41, 2, 3syl2an 273 . . 3 ((x y = A x z = A) → y = z)
54gen2 1336 . 2 yz((x y = A x z = A) → y = z)
6 eqeq1 2043 . . . 4 (y = z → (y = Az = A))
76albidv 1702 . . 3 (y = z → (x y = Ax z = A))
87eu4 1959 . 2 (∃!yx y = A ↔ (yx y = A yz((x y = A x z = A) → y = z)))
95, 8mpbiran2 847 1 (∃!yx y = Ayx y = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378  ∃!weu 1897
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-bndl 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-cleq 2030
This theorem is referenced by:  eusvnfb  4152
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