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Theorem eusv1 4570
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 1795 . . . 4  |-  ( A. x  y  =  A  ->  y  =  A )
2 sp 1795 . . . 4  |-  ( A. x  z  =  A  ->  z  =  A )
3 eqtr3 2477 . . . 4  |-  ( ( y  =  A  /\  z  =  A )  ->  y  =  z )
41, 2, 3syl2an 477 . . 3  |-  ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
54gen2 1593 . 2  |-  A. y A. z ( ( A. x  y  =  A  /\  A. x  z  =  A )  ->  y  =  z )
6 eqeq1 2453 . . . 4  |-  ( y  =  z  ->  (
y  =  A  <->  z  =  A ) )
76albidv 1680 . . 3  |-  ( y  =  z  ->  ( A. x  y  =  A 
<-> 
A. x  z  =  A ) )
87eu4 2324 . 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 910 1  |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370   E.wex 1587   E!weu 2259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-cleq 2442
This theorem is referenced by:  eusvnfb  4572
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