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

Theorem sbim 1927
 Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sbim

Proof of Theorem sbim
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbimv 1866 . . . 4
21sbbii 1739 . . 3
3 sbimv 1866 . . 3
42, 3bitri 183 . 2
5 ax-17 1507 . . 3
65sbco2vh 1919 . 2
7 ax-17 1507 . . . 4
87sbco2vh 1919 . . 3
9 ax-17 1507 . . . 4
109sbco2vh 1919 . . 3
118, 10imbi12i 238 . 2
124, 6, 113bitr3i 209 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104  wsb 1736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737 This theorem is referenced by:  sbrim  1930  sblim  1931  sbbi  1933  moimv  2066  nfraldya  2473  sbcimg  2955  zfregfr  4497  tfi  4505  peano2  4518
 Copyright terms: Public domain W3C validator