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Theorem eusv2nf 4214
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 1953 . . . 4  |-  F/ y E! y E. x  y  =  A
2 nfe1 1426 . . . . . . 7  |-  F/ x E. x  y  =  A
32nfeu 1961 . . . . . 6  |-  F/ x E! y E. x  y  =  A
4 eusv2.1 . . . . . . . . 9  |-  A  e. 
_V
54isseti 2608 . . . . . . . 8  |-  E. y 
y  =  A
6 19.8a 1523 . . . . . . . . 9  |-  ( y  =  A  ->  E. x  y  =  A )
76ancri 317 . . . . . . . 8  |-  ( y  =  A  ->  ( E. x  y  =  A  /\  y  =  A ) )
85, 7eximii 1534 . . . . . . 7  |-  E. y
( E. x  y  =  A  /\  y  =  A )
9 eupick 2021 . . . . . . 7  |-  ( ( E! y E. x  y  =  A  /\  E. y ( E. x  y  =  A  /\  y  =  A )
)  ->  ( E. x  y  =  A  ->  y  =  A ) )
108, 9mpan2 416 . . . . . 6  |-  ( E! y E. x  y  =  A  ->  ( E. x  y  =  A  ->  y  =  A ) )
113, 10alrimi 1456 . . . . 5  |-  ( E! y E. x  y  =  A  ->  A. x
( E. x  y  =  A  ->  y  =  A ) )
12 nf3 1600 . . . . 5  |-  ( F/ x  y  =  A  <->  A. x ( E. x  y  =  A  ->  y  =  A ) )
1311, 12sylibr 132 . . . 4  |-  ( E! y E. x  y  =  A  ->  F/ x  y  =  A
)
141, 13alrimi 1456 . . 3  |-  ( E! y E. x  y  =  A  ->  A. y F/ x  y  =  A )
15 dfnfc2 3627 . . . 4  |-  ( A. x  A  e.  _V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
1615, 4mpg 1381 . . 3  |-  ( F/_ x A  <->  A. y F/ x  y  =  A )
1714, 16sylibr 132 . 2  |-  ( E! y E. x  y  =  A  ->  F/_ x A )
18 eusvnfb 4212 . . . 4  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
194, 18mpbiran2 883 . . 3  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
20 eusv2i 4213 . . 3  |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
2119, 20sylbir 133 . 2  |-  ( F/_ x A  ->  E! y E. x  y  =  A )
2217, 21impbii 124 1  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   F/wnf 1390   E.wex 1422    e. wcel 1434   E!weu 1942   F/_wnfc 2207   _Vcvv 2602
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-sn 3412  df-pr 3413  df-uni 3610
This theorem is referenced by:  eusv2  4215
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