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Theorem dfnfc2 3845
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 2420 . . . 4  |-  ( F/_ x A  ->  F/_ x
y )
2 id 19 . . . 4  |-  ( F/_ x A  ->  F/_ x A )
31, 2nfeqd 2433 . . 3  |-  ( F/_ x A  ->  F/ x  y  =  A )
43alrimiv 1617 . 2  |-  ( F/_ x A  ->  A. y F/ x  y  =  A )
5 simpr 447 . . . . . 6  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  A. y F/ x  y  =  A )
6 df-nfc 2408 . . . . . . 7  |-  ( F/_ x { A }  <->  A. y F/ x  y  e.  { A } )
7 elsn 3655 . . . . . . . . 9  |-  ( y  e.  { A }  <->  y  =  A )
87nfbii 1556 . . . . . . . 8  |-  ( F/ x  y  e.  { A }  <->  F/ x  y  =  A )
98albii 1553 . . . . . . 7  |-  ( A. y F/ x  y  e. 
{ A }  <->  A. y F/ x  y  =  A )
106, 9bitri 240 . . . . . 6  |-  ( F/_ x { A }  <->  A. y F/ x  y  =  A )
115, 10sylibr 203 . . . . 5  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x { A } )
1211nfunid 3834 . . . 4  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x U. { A } )
13 nfa1 1756 . . . . . 6  |-  F/ x A. x  A  e.  V
14 nfnf1 1757 . . . . . . 7  |-  F/ x F/ x  y  =  A
1514nfal 1766 . . . . . 6  |-  F/ x A. y F/ x  y  =  A
1613, 15nfan 1771 . . . . 5  |-  F/ x
( A. x  A  e.  V  /\  A. y F/ x  y  =  A )
17 unisng 3844 . . . . . . 7  |-  ( A  e.  V  ->  U. { A }  =  A
)
1817sps 1739 . . . . . 6  |-  ( A. x  A  e.  V  ->  U. { A }  =  A )
1918adantr 451 . . . . 5  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  U. { A }  =  A
)
2016, 19nfceqdf 2418 . . . 4  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  ( F/_ x U. { A } 
<-> 
F/_ x A ) )
2112, 20mpbid 201 . . 3  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x A )
2221ex 423 . 2  |-  ( A. x  A  e.  V  ->  ( A. y F/ x  y  =  A  ->  F/_ x A ) )
234, 22impbid2 195 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 176    /\ wa 358   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   {csn 3640   U.cuni 3827
This theorem is referenced by:  eusv2nf  4532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828
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