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Theorem eusv1 4544
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 1728 . . . 4  |-  ( A. x  y  =  A  ->  y  =  A )
2 sp 1728 . . . 4  |-  ( A. x  z  =  A  ->  z  =  A )
3 eqtr3 2315 . . . 4  |-  ( ( y  =  A  /\  z  =  A )  ->  y  =  z )
41, 2, 3syl2an 463 . . 3  |-  ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
54gen2 1537 . 2  |-  A. y A. z ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
6 eqeq1 2302 . . . 4  |-  ( y  =  z  ->  (
y  =  A  <->  z  =  A ) )
76albidv 1615 . . 3  |-  ( y  =  z  ->  ( A. x  y  =  A 
<-> 
A. x  z  =  A ) )
87eu4 2195 . 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 885 1  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632   E!weu 2156
This theorem is referenced by:  eusvnfb  4546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-cleq 2289
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