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

Theorem sbn 1869
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sbn  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )

Proof of Theorem sbn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbnv 1811 . . . 4  |-  ( [ z  /  x ]  -.  ph  <->  -.  [ z  /  x ] ph )
21sbbii 1690 . . 3  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  [ y  /  z ]  -.  [ z  /  x ] ph )
3 sbnv 1811 . . 3  |-  ( [ y  /  z ]  -.  [ z  /  x ] ph  <->  -.  [ y  /  z ] [
z  /  x ] ph )
42, 3bitri 182 . 2  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  -.  [ y  /  z ] [
z  /  x ] ph )
5 ax-17 1460 . . . 4  |-  ( ph  ->  A. z ph )
65hbn 1585 . . 3  |-  ( -. 
ph  ->  A. z  -.  ph )
76sbco2v 1864 . 2  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  [ y  /  x ]  -.  ph )
85sbco2v 1864 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
98notbii 627 . 2  |-  ( -. 
[ y  /  z ] [ z  /  x ] ph  <->  -.  [ y  /  x ] ph )
104, 7, 93bitr3i 208 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbcng  2864  difab  3250  rabeq0  3291  abeq0  3292  ssfirab  6476
  Copyright terms: Public domain W3C validator