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Theorem eusv2i 4713
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 2291 . . 3  |-  F/ y E! y A. x  y  =  A
2 nfcvd 2573 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x
y )
3 eusvnf 4711 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
42, 3nfeqd 2586 . . . . 5  |-  ( E! y A. x  y  =  A  ->  F/ x  y  =  A
)
5 nf2 1889 . . . . 5  |-  ( F/ x  y  =  A  <-> 
( E. x  y  =  A  ->  A. x  y  =  A )
)
64, 5sylib 189 . . . 4  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A  ->  A. x  y  =  A ) )
7 19.2 1671 . . . 4  |-  ( A. x  y  =  A  ->  E. x  y  =  A )
86, 7impbid1 195 . . 3  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A 
<-> 
A. x  y  =  A ) )
91, 8eubid 2288 . 2  |-  ( E! y A. x  y  =  A  ->  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A ) )
109ibir 234 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 1549   E.wex 1550   F/wnf 1553    = wceq 1652   E!weu 2281
This theorem is referenced by:  eusv2nf  4714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951  df-sbc 3155  df-csb 3245
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