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Theorem eusv2nf 4712
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 2290 . . . 4  |-  F/ y E! y E. x  y  =  A
2 nfe1 1747 . . . . . . 7  |-  F/ x E. x  y  =  A
32nfeu 2296 . . . . . 6  |-  F/ x E! y E. x  y  =  A
4 eusv2.1 . . . . . . . . 9  |-  A  e. 
_V
54isseti 2954 . . . . . . . 8  |-  E. y 
y  =  A
6 19.8a 1762 . . . . . . . . 9  |-  ( y  =  A  ->  E. x  y  =  A )
76ancri 536 . . . . . . . 8  |-  ( y  =  A  ->  ( E. x  y  =  A  /\  y  =  A ) )
85, 7eximii 1587 . . . . . . 7  |-  E. y
( E. x  y  =  A  /\  y  =  A )
9 eupick 2343 . . . . . . 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 1781 . . . . 5  |-  ( E! y E. x  y  =  A  ->  A. x
( E. x  y  =  A  ->  y  =  A ) )
12 nf3 1890 . . . . 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 1781 . . 3  |-  ( E! y E. x  y  =  A  ->  A. y F/ x  y  =  A )
15 dfnfc2 4025 . . . 4  |-  ( A. x  A  e.  _V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
1615, 4mpg 1557 . . 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 4710 . . . 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 4711 . . 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 1549   E.wex 1550   F/wnf 1553    = wceq 1652    e. wcel 1725   E!weu 2280   F/_wnfc 2558   _Vcvv 2948
This theorem is referenced by:  eusv2  4713
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 2416
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-csb 3244  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008
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