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Theorem eusv2nf 4622
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 2279 . . . 4  |-  F/ y E! y E. x  y  =  A
2 nfe1 1894 . . . . . . 7  |-  F/ x E. x  y  =  A
32nfeu 2285 . . . . . 6  |-  F/ x E! y E. x  y  =  A
4 eusv2.1 . . . . . . . . 9  |-  A  e. 
_V
54isseti 3086 . . . . . . . 8  |-  E. y 
y  =  A
6 19.8a 1912 . . . . . . . . 9  |-  ( y  =  A  ->  E. x  y  =  A )
76ancri 554 . . . . . . . 8  |-  ( y  =  A  ->  ( E. x  y  =  A  /\  y  =  A ) )
85, 7eximii 1703 . . . . . . 7  |-  E. y
( E. x  y  =  A  /\  y  =  A )
9 eupick 2335 . . . . . . 7  |-  ( ( E! y E. x  y  =  A  /\  E. y ( E. x  y  =  A  /\  y  =  A )
)  ->  ( E. x  y  =  A  ->  y  =  A ) )
108, 9mpan2 675 . . . . . 6  |-  ( E! y E. x  y  =  A  ->  ( E. x  y  =  A  ->  y  =  A ) )
113, 10alrimi 1932 . . . . 5  |-  ( E! y E. x  y  =  A  ->  A. x
( E. x  y  =  A  ->  y  =  A ) )
12 nf3 2021 . . . . 5  |-  ( F/ x  y  =  A  <->  A. x ( E. x  y  =  A  ->  y  =  A ) )
1311, 12sylibr 215 . . . 4  |-  ( E! y E. x  y  =  A  ->  F/ x  y  =  A
)
141, 13alrimi 1932 . . 3  |-  ( E! y E. x  y  =  A  ->  A. y F/ x  y  =  A )
15 dfnfc2 4237 . . . 4  |-  ( A. x  A  e.  _V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
1615, 4mpg 1665 . . 3  |-  ( F/_ x A  <->  A. y F/ x  y  =  A )
1714, 16sylibr 215 . 2  |-  ( E! y E. x  y  =  A  ->  F/_ x A )
18 eusvnfb 4620 . . . 4  |-  ( E! y A. x  y  =  A  <->  ( F/_ x A  /\  A  e. 
_V ) )
194, 18mpbiran2 927 . . 3  |-  ( E! y A. x  y  =  A  <->  F/_ x A )
20 eusv2i 4621 . . 3  |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
2119, 20sylbir 216 . 2  |-  ( F/_ x A  ->  E! y E. x  y  =  A )
2217, 21impbii 190 1  |-  ( E! y E. x  y  =  A  <->  F/_ x A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657   F/wnf 1661    e. wcel 1872   E!weu 2269   F/_wnfc 2566   _Vcvv 3080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-sn 3999  df-pr 4001  df-uni 4220
This theorem is referenced by:  eusv2  4623
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