Theorem rp-fakeimass 37857

Description: A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)

Assertion

Ref	Expression
rp-fakeimass	|- ( ( ph \/ ch ) <-> ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) )

Proof of Theorem rp-fakeimass

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . . . 8 |- ( ps -> ( ph -> ps ) )
2	1	con3i 150	. . . . . . 7 |- ( -. ( ph -> ps ) -> -. ps )
3	2	pm2.21d 118	. . . . . 6 |- ( -. ( ph -> ps ) -> ( ps -> ch ) )
4	3	a1d 25	. . . . 5 |- ( -. ( ph -> ps ) -> ( ph -> ( ps -> ch ) ) )
5		ax-1 6	. . . . . 6 |- ( ch -> ( ps -> ch ) )
6	5	a1d 25	. . . . 5 |- ( ch -> ( ph -> ( ps -> ch ) ) )
7	4, 6	ja 173	. . . 4 |- ( ( ( ph -> ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) )
8		ax-2 7	. . . . 5 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
9	8	com3r 87	. . . 4 |- ( ph -> ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ch ) ) )
10	7, 9	impbid2 216	. . 3 |- ( ph -> ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) )
11		ax-1 6	. . . 4 |- ( ch -> ( ( ph -> ps ) -> ch ) )
12	11, 6	2thd 255	. . 3 |- ( ch -> ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) )
13	10, 12	jaoi 394	. 2 |- ( ( ph \/ ch ) -> ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) )
14		jarl 175	. . . . 5 |- ( ( ( ph -> ps ) -> ch ) -> ( -. ph -> ch ) )
15	14	orrd 393	. . . 4 |- ( ( ( ph -> ps ) -> ch ) -> ( ph \/ ch ) )
16	15	a1d 25	. . 3 |- ( ( ( ph -> ps ) -> ch ) -> ( ( ph -> ( ps -> ch ) ) -> ( ph \/ ch ) ) )
17		simplim 163	. . . . 5 |- ( -. ( ph -> ( ps -> ch ) ) -> ph )
18	17	orcd 407	. . . 4 |- ( -. ( ph -> ( ps -> ch ) ) -> ( ph \/ ch ) )
19	18	a1i 11	. . 3 |- ( -. ( ( ph -> ps ) -> ch ) -> ( -. ( ph -> ( ps -> ch ) ) -> ( ph \/ ch ) ) )
20	16, 19	bija 370	. 2 |- ( ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) -> ( ph \/ ch ) )
21	13, 20	impbii 199	1 |- ( ( ph \/ ch ) <-> ( ( ( ph -> ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383

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-or 385

This theorem is referenced by: (None)