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Theorem eusv2nf 4662
Description: Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1  |-  A  e. 
_V
Assertion
Ref Expression
eusv2nf  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
Distinct variable groups:    x, y    y, A
Allowed substitution hint:    A( x)

Proof of Theorem eusv2nf
StepHypRef Expression
1 nfeu1 2249 . . . 4  |-  F/ y E! y E. x  y  =  A
2 nfe1 1739 . . . . . . 7  |-  F/ x E. x  y  =  A
32nfeu 2255 . . . . . 6  |-  F/ x E! y E. x  y  =  A
4 eusv2.1 . . . . . . . . 9  |-  A  e. 
_V
54isseti 2906 . . . . . . . 8  |-  E. y 
y  =  A
6 19.8a 1754 . . . . . . . . 9  |-  ( y  =  A  ->  E. x  y  =  A )
76ancri 536 . . . . . . . 8  |-  ( y  =  A  ->  ( E. x  y  =  A  /\  y  =  A ) )
85, 7eximii 1584 . . . . . . 7  |-  E. y
( E. x  y  =  A  /\  y  =  A )
9 eupick 2302 . . . . . . 7  |-  ( ( E! y E. x  y  =  A  /\  E. y ( E. x  y  =  A  /\  y  =  A )
)  ->  ( E. x  y  =  A  ->  y  =  A ) )
108, 9mpan2 653 . . . . . 6  |-  ( E! y E. x  y  =  A  ->  ( E. x  y  =  A  ->  y  =  A ) )
113, 10alrimi 1773 . . . . 5  |-  ( E! y E. x  y  =  A  ->  A. x
( E. x  y  =  A  ->  y  =  A ) )
12 nf3 1879 . . . . 5  |-  ( F/ x  y  =  A  <->  A. x ( E. x  y  =  A  ->  y  =  A ) )
1311, 12sylibr 204 . . . 4  |-  ( E! y E. x  y  =  A  ->  F/ x  y  =  A
)
141, 13alrimi 1773 . . 3  |-  ( E! y E. x  y  =  A  ->  A. y F/ x  y  =  A )
15 dfnfc2 3976 . . . 4  |-  ( A. x  A  e.  _V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
1615, 4mpg 1554 . . 3  |-  ( F/_ x A  <->  A. y F/ x  y  =  A )
1714, 16sylibr 204 . 2  |-  ( E! y E. x  y  =  A  ->  F/_ x A )
18 eusvnfb 4660 . . . 4  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
194, 18mpbiran2 886 . . 3  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
20 eusv2i 4661 . . 3  |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
2119, 20sylbir 205 . 2  |-  ( F/_ x A  ->  E! y E. x  y  =  A )
2217, 21impbii 181 1  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   F/wnf 1550    = wceq 1649    e. wcel 1717   E!weu 2239   F/_wnfc 2511   _Vcvv 2900
This theorem is referenced by:  eusv2  4663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-v 2902  df-sbc 3106  df-csb 3196  df-un 3269  df-sn 3764  df-pr 3765  df-uni 3959
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