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Theorem eusv1 4130
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 1378 . . . 4 (x y = Ay = A)
2 sp 1378 . . . 4 (x z = Az = A)
3 eqtr3 2037 . . . 4 ((y = A z = A) → y = z)
41, 2, 3syl2an 273 . . 3 ((x y = A x z = A) → y = z)
54gen2 1315 . 2 yz((x y = A x z = A) → y = z)
6 eqeq1 2024 . . . 4 (y = z → (y = Az = A))
76albidv 1683 . . 3 (y = z → (x y = Ax z = A))
87eu4 1940 . 2 (∃!yx y = A ↔ (yx y = A yz((x y = A x z = A) → y = z)))
95, 8mpbiran2 834 1 (∃!yx y = Ayx y = A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224   = wceq 1226  wex 1358  ∃!weu 1878
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-cleq 2011
This theorem is referenced by:  eusvnfb  4132
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