Theorem condc 855

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

Step	Hyp	Ref	Expression
1		dcstab 846	. 2 |- (DECID ph -> STAB ph )
2		const 854	. 2 |- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
3	1, 2	syl 14	1 |- (DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4  STAB wstab 832  DECID wdc 836

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 711

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837

This theorem is referenced by:  pm2.18dc  857  con1dc  858  con4biddc  859  pm2.521gdc  870  pm2.521dcALT  872  con34bdc  873  necon4aidc  2445  necon4addc  2447  necon4bddc  2448  necon4ddc  2449  nn0n0n1ge2b  9482  gcdeq0  12383  lcmeq0  12478  pcdvdsb  12728  pc2dvds  12738  pcfac  12758  infpnlem1  12767  m1lgs  15647