Theorem con3th 573

Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This version of con3 86 demonstrates the use of the weak deduction theorem to derive it from con3i 90.

Assertion

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

Proof of Theorem con3th

Step	Hyp	Ref	Expression
1		id 9	. . . 4 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> (ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))))
2	1	negbid 463	. . 3 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> (-. ps <-> -. ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))))
3	2	imbi1d 465	. 2 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> ((-. ps -> -. ph) <-> (-. ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))) -> -. ph)))
4	1	imbi2d 464	. . . 4 |- ((ps <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> ((ph -> ps) <-> (ph -> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))))))
5		id 9	. . . . 5 |- ((ph <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> (ph <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))))
6	5	imbi2d 464	. . . 4 |- ((ph <-> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps)))) -> ((ph -> ph) <-> (ph -> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))))))
7		id 9	. . . 4 |- (ph -> ph)
8	4, 6, 7	elimh 571	. . 3 |- (ph -> ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))))
9	8	con3i 90	. 2 |- (-. ((ps /\ (ph -> ps)) \/ (ph /\ -. (ph -> ps))) -> -. ph)
10	3, 9	dedt 572	1 |- ((ph -> ps) -> (-. ps -> -. ph))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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