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Theorem eusv2i 4346
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 1988 . . 3  |-  F/ y E! y A. x  y  =  A
2 nfcvd 2259 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x
y )
3 eusvnf 4344 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
42, 3nfeqd 2273 . . . . 5  |-  ( E! y A. x  y  =  A  ->  F/ x  y  =  A
)
5 nf2 1631 . . . . 5  |-  ( F/ x  y  =  A  <-> 
( E. x  y  =  A  ->  A. x  y  =  A )
)
64, 5sylib 121 . . . 4  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A  ->  A. x  y  =  A ) )
7 19.2 1602 . . . 4  |-  ( A. x  y  =  A  ->  E. x  y  =  A )
86, 7impbid1 141 . . 3  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A 
<-> 
A. x  y  =  A ) )
91, 8eubid 1984 . 2  |-  ( E! y A. x  y  =  A  ->  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A ) )
109ibir 176 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 1314    = wceq 1316   F/wnf 1421   E.wex 1453   E!weu 1977
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976
This theorem is referenced by:  eusv2nf  4347
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