Theorem condc 748

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 742	. 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 35	. . . 4 |- (-. ph -> ((-. ph -> -. ps) -> -. ps))
5		ax-in2 545	. . . 4 |- (-. ps -> (ps -> ph))
6	4, 5	syl6 29	. . 3 |- (-. ph -> ((-. ph -> -. ps) -> (ps -> ph)))
7	3, 6	jaoi 635	. 2 |- ((ph \/ -. ph) -> ((-. ph -> -. ps) -> (ps -> ph)))
8	1, 7	sylbi 114	1 |- (DECID ph -> ((-. ph -> -. ps) -> (ps -> ph)))

Syntax hints:  -. wn 3   -> wi 4   \/ wo 628  DECID wdc 741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm2.18dc  749  con1dc  752  con4biddc  753  pm2.521dc  763  con34bdc  764  necon4aidc  2267  necon4addc  2269  necon4bddc  2270  necon4ddc  2271  nn0n0n1ge2b  8096