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Theorem eusv1 4184
Description: Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 14-Oct-2010.)
Assertion
Ref Expression
eusv1  |-  ( E! y A. x  y  =  A  <->  E. y A. x  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 1401 . . . 4  |-  ( A. x  y  =  A  ->  y  =  A )
2 sp 1401 . . . 4  |-  ( A. x  z  =  A  ->  z  =  A )
3 eqtr3 2059 . . . 4  |-  ( ( y  =  A  /\  z  =  A )  ->  y  =  z )
41, 2, 3syl2an 273 . . 3  |-  ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
54gen2 1339 . 2  |-  A. y A. z ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
6 eqeq1 2046 . . . 4  |-  ( y  =  z  ->  (
y  =  A  <->  z  =  A ) )
76albidv 1705 . . 3  |-  ( y  =  z  ->  ( A. x  y  =  A 
<-> 
A. x  z  =  A ) )
87eu4 1962 . 2  |-  ( E! y A. x  y  =  A  <->  ( E. y A. x  y  =  A  /\  A. y A. z ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z ) ) )
95, 8mpbiran2 848 1  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243   E.wex 1381   E!weu 1900
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033
This theorem is referenced by:  eusvnfb  4186
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