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

Theorem condc 854
Description: Contraposition of a decidable proposition.

This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning.

(Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)

Assertion
Ref Expression
condc  |-  (DECID  ph  ->  ( ( -.  ph  ->  -. 
ps )  ->  ( ps  ->  ph ) ) )

Proof of Theorem condc
StepHypRef Expression
1 dcstab 845 . 2  |-  (DECID  ph  -> STAB  ph )
2 const 853 . 2  |-  (STAB  ph  ->  ( ( -.  ph  ->  -. 
ps )  ->  ( ps  ->  ph ) ) )
31, 2syl 14 1  |-  (DECID  ph  ->  ( ( -.  ph  ->  -. 
ps )  ->  ( ps  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  STAB wstab 831  DECID wdc 835
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
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836
This theorem is referenced by:  pm2.18dc  856  con1dc  857  con4biddc  858  pm2.521gdc  869  pm2.521dcALT  871  con34bdc  872  necon4aidc  2428  necon4addc  2430  necon4bddc  2431  necon4ddc  2432  nn0n0n1ge2b  9363  gcdeq0  12013  lcmeq0  12106  pcdvdsb  12355  pc2dvds  12365  pcfac  12385  infpnlem1  12394  m1lgs  14930
  Copyright terms: Public domain W3C validator