Theorem condc 861

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 852	. 2 |- (DECID ph -> STAB ph )
2		const 860	. 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 838  DECID wdc 842

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 619  ax-in2 620  ax-io 717

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843

This theorem is referenced by:  pm2.18dc  863  con1dc  864  con4biddc  865  pm2.521gdc  876  pm2.521dcALT  878  con34bdc  879  necon4aidc  2480  necon4addc  2482  necon4bddc  2483  necon4ddc  2484  nn0n0n1ge2b  9657  gcdeq0  12673  lcmeq0  12768  pcdvdsb  13018  pc2dvds  13028  pcfac  13048  infpnlem1  13057  m1lgs  15958  exmidcon  16780