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Theorem eusv1 4210
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 1442 . . . 4  |-  ( A. x  y  =  A  ->  y  =  A )
2 sp 1442 . . . 4  |-  ( A. x  z  =  A  ->  z  =  A )
3 eqtr3 2101 . . . 4  |-  ( ( y  =  A  /\  z  =  A )  ->  y  =  z )
41, 2, 3syl2an 283 . . 3  |-  ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
54gen2 1380 . 2  |-  A. y A. z ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
6 eqeq1 2088 . . . 4  |-  ( y  =  z  ->  (
y  =  A  <->  z  =  A ) )
76albidv 1746 . . 3  |-  ( y  =  z  ->  ( A. x  y  =  A 
<-> 
A. x  z  =  A ) )
87eu4 2004 . 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 883 1  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422   E!weu 1942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-cleq 2075
This theorem is referenced by:  eusvnfb  4212
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