MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nf4 Unicode version

Theorem nf4 1880
Description: Variable  x is effectively not free in  ph iff  ph is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
nf4  |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )

Proof of Theorem nf4
StepHypRef Expression
1 nf2 1878 . 2  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
2 imor 402 . 2  |-  ( ( E. x ph  ->  A. x ph )  <->  ( -.  E. x ph  \/  A. x ph ) )
3 orcom 377 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
4 alnex 1549 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
54orbi2i 506 . . 3  |-  ( ( A. x ph  \/  A. x  -.  ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
63, 5bitr4i 244 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  A. x  -.  ph )
)
71, 2, 63bitri 263 1  |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358   A.wal 1546   E.wex 1547   F/wnf 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-or 360  df-ex 1548  df-nf 1551
  Copyright terms: Public domain W3C validator