Theorem con3th 924

Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 126 demonstrates the use of the weak deduction theorem dedt 923 to derive it from con3i 127. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.)

Assertion

Ref	Expression
con3th	|- ((ph -> ps) -> (-. ps -> -. ph))

Proof of Theorem con3th

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> (ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))))
2	1	notbid 285	. . 3 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> (-. ps <-> -. ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))))
3	2	imbi1d 308	. 2 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> ((-. ps -> -. ph) <-> (-. ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))) -> -. ph)))
4	1	imbi2d 307	. . . 4 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> ((ph -> ps) <-> (ph -> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))))))
5		id 19	. . . . 5 |- ((ph <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> (ph <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))))
6	5	imbi2d 307	. . . 4 |- ((ph <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> ((ph -> ph) <-> (ph -> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))))))
7		id 19	. . . 4 |- (ph -> ph)
8	4, 6, 7	elimh 922	. . 3 |- (ph -> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))))
9	8	con3i 127	. 2 |- (-. ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))) -> -. ph)
10	3, 9	dedt 923	1 |- ((ph -> ps) -> (-. ps -> -. ph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is referenced by: (None)