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Theorem eusv1 4373
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 1488 . . . 4  |-  ( A. x  y  =  A  ->  y  =  A )
2 sp 1488 . . . 4  |-  ( A. x  z  =  A  ->  z  =  A )
3 eqtr3 2159 . . . 4  |-  ( ( y  =  A  /\  z  =  A )  ->  y  =  z )
41, 2, 3syl2an 287 . . 3  |-  ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
54gen2 1426 . 2  |-  A. y A. z ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
6 eqeq1 2146 . . . 4  |-  ( y  =  z  ->  (
y  =  A  <->  z  =  A ) )
76albidv 1796 . . 3  |-  ( y  =  z  ->  ( A. x  y  =  A 
<-> 
A. x  z  =  A ) )
87eu4 2061 . 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 925 1  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468   E!weu 1999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132
This theorem is referenced by:  eusvnfb  4375
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