Theorem confun5 41110

Description: An attempt at derivative. Resisted simplest path to a proof. Interesting that ch, th, ta, et were all provable. (Contributed by Jarvin Udandy, 7-Sep-2020.)

Hypotheses

Ref	Expression
confun5.1	|- ph
confun5.2	|- ( ( ph -> ps ) -> ps )
confun5.3	|- ( ps -> ( ph -> ch ) )
confun5.4	|- ( ( ch -> th ) -> ( ( ph -> th ) <-> ps ) )
confun5.5	|- ( ta <-> ( ch -> th ) )
confun5.6	|- ( et <-> -. ( ch -> ( ch /\ -. ch ) ) )
confun5.7	|- ps
confun5.8	|- ( ch -> th )

Assertion

Ref	Expression
confun5	|- ( ch -> ( et <-> ta ) )

Proof of Theorem confun5

Step	Hyp	Ref	Expression
1		confun5.1	. . . . . 6 |- ph
2		confun5.7	. . . . . . 7 |- ps
3		confun5.3	. . . . . . 7 |- ( ps -> ( ph -> ch ) )
4	2, 3	ax-mp 5	. . . . . 6 |- ( ph -> ch )
5	1, 4	ax-mp 5	. . . . 5 |- ch
6	5	atnaiana 41090	. . . 4 |- -. ( ch -> ( ch /\ -. ch ) )
7		confun5.6	. . . . . 6 |- ( et <-> -. ( ch -> ( ch /\ -. ch ) ) )
8		bicom1 211	. . . . . 6 |- ( ( et <-> -. ( ch -> ( ch /\ -. ch ) ) ) -> ( -. ( ch -> ( ch /\ -. ch ) ) <-> et ) )
9	7, 8	ax-mp 5	. . . . 5 |- ( -. ( ch -> ( ch /\ -. ch ) ) <-> et )
10	9	biimpi 206	. . . 4 |- ( -. ( ch -> ( ch /\ -. ch ) ) -> et )
11	6, 10	ax-mp 5	. . 3 |- et
12		confun5.8	. . . 4 |- ( ch -> th )
13		confun5.5	. . . . . 6 |- ( ta <-> ( ch -> th ) )
14		bicom1 211	. . . . . 6 |- ( ( ta <-> ( ch -> th ) ) -> ( ( ch -> th ) <-> ta ) )
15	13, 14	ax-mp 5	. . . . 5 |- ( ( ch -> th ) <-> ta )
16	15	biimpi 206	. . . 4 |- ( ( ch -> th ) -> ta )
17	12, 16	ax-mp 5	. . 3 |- ta
18	11, 17	2th 254	. 2 |- ( et <-> ta )
19		ax-1 6	. 2 |- ( ( et <-> ta ) -> ( ch -> ( et <-> ta ) ) )
20	18, 19	ax-mp 5	1 |- ( ch -> ( et <-> ta ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

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-an 386  df-tru 1486  df-fal 1489

This theorem is referenced by: (None)