ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nf4r Unicode version

Theorem nf4r 1682
Description: If  ph is always true or always false, then variable  x is effectively not free in 
ph. The converse holds given a decidability condition, as seen at nf4dc 1681. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
nf4r  |-  ( ( A. x ph  \/  A. x  -.  ph )  ->  F/ x ph )

Proof of Theorem nf4r
StepHypRef Expression
1 orcom 729 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
2 alnex 1510 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
32orbi2i 763 . . 3  |-  ( ( A. x ph  \/  A. x  -.  ph )  <->  ( A. x ph  \/  -.  E. x ph )
)
41, 3bitr4i 187 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  <->  ( A. x ph  \/  A. x  -.  ph )
)
5 imorr 722 . . 3  |-  ( ( -.  E. x ph  \/  A. x ph )  ->  ( E. x ph  ->  A. x ph )
)
6 nf2 1679 . . 3  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
75, 6sylibr 134 . 2  |-  ( ( -.  E. x ph  \/  A. x ph )  ->  F/ x ph )
84, 7sylbir 135 1  |-  ( ( A. x ph  \/  A. x  -.  ph )  ->  F/ x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 709   A.wal 1362   F/wnf 1471   E.wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator