Theorem tbw-bijust 1623

Description: Justification for tbw-negdf 1624. (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 203	. 2 |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
2		pm2.21 120	. . . . 5 |- ( -. ( ps -> ph ) -> ( ( ps -> ph ) -> F. ) )
3	2	imim2i 16	. . . 4 |- ( ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
4		id 22	. . . . . 6 |- ( -. ( ps -> ph ) -> -. ( ps -> ph ) )
5		falim 1498	. . . . . 6 |- ( F. -> -. ( ps -> ph ) )
6	4, 5	ja 173	. . . . 5 |- ( ( ( ps -> ph ) -> F. ) -> -. ( ps -> ph ) )
7	6	imim2i 16	. . . 4 |- ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
8	3, 7	impbii 199	. . 3 |- ( ( ( ph -> ps ) -> -. ( ps -> ph ) ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
9	8	notbii 310	. 2 |- ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) <-> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
10		pm2.21 120	. . 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 1498	. . . . 5 |- ( F. -> ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) ) )
13	11, 12	ja 173	. . . 4 |- ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) ) )
14	13	pm2.43i 52	. . 3 |- ( ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) -> -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) )
15	10, 14	impbii 199	. 2 |- ( -. ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) <-> ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) )
16	1, 9, 15	3bitri 286	1 |- ( ( ph <-> ps ) <-> ( ( ( ph -> ps ) -> ( ( ps -> ph ) -> F. ) ) -> F. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   F. wfal 1488

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 197  df-tru 1486  df-fal 1489

This theorem is referenced by:  tbw-negdf  1624