Theorem dcfromcon 1494

Description: The decidability of a proposition ch follows from a suitable instance of the principle of contraposition. Therefore, if we were to introduce contraposition as a general principle (without the decidability condition in condc 861), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since the principle of contraposition is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)

Hypotheses

Ref	Expression
dcfromcon.1	|- ( ph <-> ( ch \/ -. ch ) )
dcfromcon.2	|- ( ps <-> T. )
dcfromcon.3	|- ( ( -. ph -> -. ps ) -> ( ps -> ph ) )

Assertion

Ref	Expression
dcfromcon	|- DECID ch

Proof of Theorem dcfromcon

Step	Hyp	Ref	Expression
1		nnexmid 858	. . . . 5 |- -. -. ( ch \/ -. ch )
2	1	pm2.21i 651	. . . 4 |- ( -. ( ch \/ -. ch ) -> -. T. )
3		dcfromcon.3	. . . . 5 |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) )
4		dcfromcon.1	. . . . . . 7 |- ( ph <-> ( ch \/ -. ch ) )
5	4	notbii 674	. . . . . 6 |- ( -. ph <-> -. ( ch \/ -. ch ) )
6		dcfromcon.2	. . . . . . 7 |- ( ps <-> T. )
7	6	notbii 674	. . . . . 6 |- ( -. ps <-> -. T. )
8	5, 7	imbi12i 239	. . . . 5 |- ( ( -. ph -> -. ps ) <-> ( -. ( ch \/ -. ch ) -> -. T. ) )
9	6, 4	imbi12i 239	. . . . 5 |- ( ( ps -> ph ) <-> ( T. -> ( ch \/ -. ch ) ) )
10	3, 8, 9	3imtr3i 200	. . . 4 |- ( ( -. ( ch \/ -. ch ) -> -. T. ) -> ( T. -> ( ch \/ -. ch ) ) )
11	2, 10	ax-mp 5	. . 3 |- ( T. -> ( ch \/ -. ch ) )
12	11	mptru 1407	. 2 |- ( ch \/ -. ch )
13		df-dc 843	. 2 |- (DECID ch <-> ( ch \/ -. ch ) )
14	12, 13	mpbir 146	1 |- DECID ch

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 716  DECID wdc 842   T. wtru 1399

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-dc 843  df-tru 1401

This theorem is referenced by: (None)