Theorem tbw-bijust 1463

Description: Justification for tbw-negdf 1464. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem tbw-bijust

Step	Hyp	Ref	Expression
1		dfbi1 184	. 2 |- ((ph <-> ps) <-> -. ((ph -> ps) -> -. (ps -> ph)))
2		pm2.21 100	. . . . 5 |- (-. (ps -> ph) -> ((ps -> ph) -> F. ))
3	2	imim2i 13	. . . 4 |- (((ph -> ps) -> -. (ps -> ph)) -> ((ph -> ps) -> ((ps -> ph) -> F. )))
4		id 19	. . . . . 6 |- (-. (ps -> ph) -> -. (ps -> ph))
5		falim 1328	. . . . . 6 |- ( F. -> -. (ps -> ph))
6	4, 5	ja 153	. . . . 5 |- (((ps -> ph) -> F. ) -> -. (ps -> ph))
7	6	imim2i 13	. . . 4 |- (((ph -> ps) -> ((ps -> ph) -> F. )) -> ((ph -> ps) -> -. (ps -> ph)))
8	3, 7	impbii 180	. . 3 |- (((ph -> ps) -> -. (ps -> ph)) <-> ((ph -> ps) -> ((ps -> ph) -> F. )))
9	8	notbii 287	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) <-> -. ((ph -> ps) -> ((ps -> ph) -> F. )))
10		pm2.21 100	. . 3 |- (-. ((ph -> ps) -> ((ps -> ph) -> F. )) -> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))
11		ax-1 6	. . . . 5 |- (-. ((ph -> ps) -> ((ps -> ph) -> F. )) -> ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. ))))
12		falim 1328	. . . . 5 |- ( F. -> ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. ))))
13	11, 12	ja 153	. . . 4 |- ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. ))))
14	13	pm2.43i 43	. . 3 |- ((((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ) -> -. ((ph -> ps) -> ((ps -> ph) -> F. )))
15	10, 14	impbii 180	. 2 |- (-. ((ph -> ps) -> ((ps -> ph) -> F. )) <-> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))
16	1, 9, 15	3bitri 262	1 |- ((ph <-> ps) <-> (((ph -> ps) -> ((ps -> ph) -> F. )) -> F. ))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   F. wfal 1317

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-tru 1319  df-fal 1320

This theorem is referenced by:  tbw-negdf  1464