MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnfc2 Structured version   Unicode version

Theorem dfnfc2 4237
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 2581 . . . 4  |-  ( F/_ x A  ->  F/_ x
y )
2 id 22 . . . 4  |-  ( F/_ x A  ->  F/_ x A )
31, 2nfeqd 2587 . . 3  |-  ( F/_ x A  ->  F/ x  y  =  A )
43alrimiv 1767 . 2  |-  ( F/_ x A  ->  A. y F/ x  y  =  A )
5 simpr 462 . . . . . 6  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  A. y F/ x  y  =  A )
6 df-nfc 2568 . . . . . . 7  |-  ( F/_ x { A }  <->  A. y F/ x  y  e.  { A } )
7 elsn 4012 . . . . . . . . 9  |-  ( y  e.  { A }  <->  y  =  A )
87nfbii 1689 . . . . . . . 8  |-  ( F/ x  y  e.  { A }  <->  F/ x  y  =  A )
98albii 1685 . . . . . . 7  |-  ( A. y F/ x  y  e. 
{ A }  <->  A. y F/ x  y  =  A )
106, 9bitri 252 . . . . . 6  |-  ( F/_ x { A }  <->  A. y F/ x  y  =  A )
115, 10sylibr 215 . . . . 5  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x { A } )
1211nfunid 4226 . . . 4  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x U. { A } )
13 nfa1 1956 . . . . . 6  |-  F/ x A. x  A  e.  V
14 nfnf1 1958 . . . . . . 7  |-  F/ x F/ x  y  =  A
1514nfal 2007 . . . . . 6  |-  F/ x A. y F/ x  y  =  A
1613, 15nfan 1988 . . . . 5  |-  F/ x
( A. x  A  e.  V  /\  A. y F/ x  y  =  A )
17 unisng 4235 . . . . . . 7  |-  ( A  e.  V  ->  U. { A }  =  A
)
1817sps 1920 . . . . . 6  |-  ( A. x  A  e.  V  ->  U. { A }  =  A )
1918adantr 466 . . . . 5  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  U. { A }  =  A
)
2016, 19nfceqdf 2575 . . . 4  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  ( F/_ x U. { A } 
<-> 
F/_ x A ) )
2112, 20mpbid 213 . . 3  |-  ( ( A. x  A  e.  V  /\  A. y F/ x  y  =  A )  ->  F/_ x A )
2221ex 435 . 2  |-  ( A. x  A  e.  V  ->  ( A. y F/ x  y  =  A  ->  F/_ x A ) )
234, 22impbid2 207 1  |-  ( A. x  A  e.  V  ->  ( F/_ x A  <->  A. y F/ x  y  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   F/wnf 1661    e. wcel 1872   F/_wnfc 2566   {csn 3998   U.cuni 4219
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-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220
This theorem is referenced by:  eusv2nf  4622
  Copyright terms: Public domain W3C validator