Theorem tbw-bijust 14165

Description: Justification for tbw-negdf 14166.

Assertion

Ref	Expression
tbw-bijust	|- ((ph <-> ps) <-> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))

Proof of Theorem tbw-bijust

Step	Hyp	Ref	Expression
1		dfbi1 175	. 2 |- ((ph <-> ps) <-> -. ((ph -> ps) -> -. (ps -> ph)))
2		pm2.21 92	. . . . 5 |- (-. (ps -> ph) -> ((ps -> ph) -> F. ))
3	2	imim2i 11	. . . 4 |- (((ph -> ps) -> -. (ps -> ph)) -> ((ph -> ps) -> ((ps -> ph) -> F. )))
4		id 73	. . . . . 6 |- (-. (ps -> ph) -> -. (ps -> ph))
5		FiA 14104	. . . . . 6 |- ( F. -> -. (ps -> ph))
6	4, 5	ja 152	. . . . 5 |- (((ps -> ph) -> F. ) -> -. (ps -> ph))
7	6	imim2i 11	. . . 4 |- (((ph -> ps) -> ((ps -> ph) -> F. )) -> ((ph -> ps) -> -. (ps -> ph)))
8	3, 7	impbii 174	. . 3 |- (((ph -> ps) -> -. (ps -> ph)) <-> ((ph -> ps) -> ((ps -> ph) -> F. )))
9	8	notbii 204	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) <-> -. ((ph -> ps) -> ((ps -> ph) -> F. )))
10		pm2.21 92	. . 3 |- (-. ((ph -> ps) -> ((ps -> ph) -> F. )) -> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))
11		ax-1 4	. . . . 5 |- (-. ((ph -> ps) -> ((ps -> ph) -> F. )) -> ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. ))))
12		FiA 14104	. . . . 5 |- ( F. -> ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. ))))
13	11, 12	ja 152	. . . 4 |- ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. ))))
14	13	pm2.43i 78	. . 3 |- ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. )))
15	10, 14	impbii 174	. 2 |- (-. ((ph -> ps) -> ((ps -> ph) -> F. )) <-> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))
16	1, 9, 15	3bitri 194	1 |- ((ph <-> ps) <-> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   F. wfal 1261

This theorem is referenced by:  tbw-negdf 14166

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-tru 1262  df-fal 1263

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