Theorem condc 787

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.)

Assertion

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

Proof of Theorem condc

Step	Hyp	Ref	Expression
1		df-dc 781	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		ax-1 5	. . . 4 |- ( ph -> ( ps -> ph ) )
3	2	a1d 22	. . 3 |- ( ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
4		pm2.27 39	. . . 4 |- ( -. ph -> ( ( -. ph -> -. ps ) -> -. ps ) )
5		ax-in2 580	. . . 4 |- ( -. ps -> ( ps -> ph ) )
6	4, 5	syl6 33	. . 3 |- ( -. ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
7	3, 6	jaoi 671	. 2 |- ( ( ph \/ -. ph ) -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
8	1, 7	sylbi 119	1 |- (DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 664  DECID wdc 780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781

This theorem is referenced by:  pm2.18dc  788  con1dc  791  con4biddc  792  pm2.521dc  802  con34bdc  803  necon4aidc  2323  necon4addc  2325  necon4bddc  2326  necon4ddc  2327  nn0n0n1ge2b  8796  gcdeq0  11050  lcmeq0  11135