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

Theorem sbn 1823
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] ¬ φ ↔ ¬ [y / x]φ)

Proof of Theorem sbn
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 sbnv 1765 . . . 4 ([z / x] ¬ φ ↔ ¬ [z / x]φ)
21sbbii 1645 . . 3 ([y / z][z / x] ¬ φ ↔ [y / z] ¬ [z / x]φ)
3 sbnv 1765 . . 3 ([y / z] ¬ [z / x]φ ↔ ¬ [y / z][z / x]φ)
42, 3bitri 173 . 2 ([y / z][z / x] ¬ φ ↔ ¬ [y / z][z / x]φ)
5 ax-17 1416 . . . 4 (φzφ)
65hbn 1541 . . 3 φz ¬ φ)
76sbco2v 1818 . 2 ([y / z][z / x] ¬ φ ↔ [y / x] ¬ φ)
85sbco2v 1818 . . 3 ([y / z][z / x]φ ↔ [y / x]φ)
98notbii 593 . 2 (¬ [y / z][z / x]φ ↔ ¬ [y / x]φ)
104, 7, 93bitr3i 199 1 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  [wsb 1642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbcng  2797  difab  3200  rabeq0  3241  abeq0  3242
  Copyright terms: Public domain W3C validator