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Theorem eusv2i 4590
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 2273 . . 3  |-  F/ y E! y A. x  y  =  A
2 nfcvd 2614 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x
y )
3 eusvnf 4588 . . . . . 6  |-  ( E! y A. x  y  =  A  ->  F/_ x A )
42, 3nfeqd 2620 . . . . 5  |-  ( E! y A. x  y  =  A  ->  F/ x  y  =  A
)
5 nf2 1897 . . . . 5  |-  ( F/ x  y  =  A  <-> 
( E. x  y  =  A  ->  A. x  y  =  A )
)
64, 5sylib 196 . . . 4  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A  ->  A. x  y  =  A ) )
7 19.2 1714 . . . 4  |-  ( A. x  y  =  A  ->  E. x  y  =  A )
86, 7impbid1 203 . . 3  |-  ( E! y A. x  y  =  A  ->  ( E. x  y  =  A 
<-> 
A. x  y  =  A ) )
91, 8eubid 2281 . 2  |-  ( E! y A. x  y  =  A  ->  ( E! y E. x  y  =  A  <->  E! y A. x  y  =  A ) )
109ibir 242 1  |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368    = wceq 1370   E.wex 1587   F/wnf 1590   E!weu 2260
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 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739
This theorem is referenced by:  eusv2nf  4591
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