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Theorem eusv2i 4251
Description: Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
eusv2i  |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem eusv2i
StepHypRef Expression
1 nfeu1 1956 . . 3  |-  F/ y E! y A. x  y  =  A
2 nfcvd 2226 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x
y )
3 eusvnf 4249 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
42, 3nfeqd 2239 . . . . 5  |-  ( E! y A. x  y  =  A  ->  F/ x  y  =  A
)
5 nf2 1601 . . . . 5  |-  ( F/ x  y  =  A  <-> 
( E. x  y  =  A  ->  A. x  y  =  A )
)
64, 5sylib 120 . . . 4  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A  ->  A. x  y  =  A ) )
7 19.2 1572 . . . 4  |-  ( A. x  y  =  A  ->  E. x  y  =  A )
86, 7impbid1 140 . . 3  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A 
<-> 
A. x  y  =  A ) )
91, 8eubid 1952 . 2  |-  ( E! y A. x  y  =  A  ->  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A ) )
109ibir 175 1  |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1285    = wceq 1287   F/wnf 1392   E.wex 1424   E!weu 1945
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923
This theorem is referenced by:  eusv2nf  4252
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