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Theorem dfnfc2 4033
Description: An alternative statement of the effective freeness of a class  A, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
dfnfc2  |-  ( A. x  A  e.  V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    A( x)    V( x, y)

Proof of Theorem dfnfc2
StepHypRef Expression
1 nfcvd 2573 . . . 4  |-  ( F/_ x A  ->  F/_ x
y )
2 id 20 . . . 4  |-  ( F/_ x A  ->  F/_ x A )
31, 2nfeqd 2586 . . 3  |-  ( F/_ x A  ->  F/ x  y  =  A )
43alrimiv 1641 . 2  |-  ( F/_ x A  ->  A. y F/ x  y  =  A )
5 simpr 448 . . . . . 6  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  A. y F/ x  y  =  A )
6 df-nfc 2561 . . . . . . 7  |-  ( F/_ x { A }  <->  A. y F/ x  y  e.  { A } )
7 elsn 3829 . . . . . . . . 9  |-  ( y  e.  { A }  <->  y  =  A )
87nfbii 1578 . . . . . . . 8  |-  ( F/ x  y  e.  { A }  <->  F/ x  y  =  A )
98albii 1575 . . . . . . 7  |-  ( A. y F/ x  y  e. 
{ A }  <->  A. y F/ x  y  =  A )
106, 9bitri 241 . . . . . 6  |-  ( F/_ x { A }  <->  A. y F/ x  y  =  A )
115, 10sylibr 204 . . . . 5  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x { A } )
1211nfunid 4022 . . . 4  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x U. { A } )
13 nfa1 1806 . . . . . 6  |-  F/ x A. x  A  e.  V
14 nfnf1 1808 . . . . . . 7  |-  F/ x F/ x  y  =  A
1514nfal 1864 . . . . . 6  |-  F/ x A. y F/ x  y  =  A
1613, 15nfan 1846 . . . . 5  |-  F/ x
( A. x  A  e.  V  /\  A. y F/ x  y  =  A )
17 unisng 4032 . . . . . . 7  |-  ( A  e.  V  ->  U. { A }  =  A
)
1817sps 1770 . . . . . 6  |-  ( A. x  A  e.  V  ->  U. { A }  =  A )
1918adantr 452 . . . . 5  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  U. { A }  =  A
)
2016, 19nfceqdf 2571 . . . 4  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  ( F/_ x U. { A } 
<-> 
F/_ x A ) )
2112, 20mpbid 202 . . 3  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x A )
2221ex 424 . 2  |-  ( A. x  A  e.  V  ->  ( A. y F/ x  y  =  A  ->  F/_ x A ) )
234, 22impbid2 196 1  |-  ( A. x  A  e.  V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2559   {csn 3814   U.cuni 4015
This theorem is referenced by:  eusv2nf  4721
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 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821  df-uni 4016
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