Theorem confun4

Description: An attempt at derivative. Resisted simplest path to a proof. (Contributed by Jarvin Udandy, 6-Sep-2020)

	Ref	Expression
Hypotheses	confun4.1	|- ph
	confun4.2	|- ( ( ph -> ps ) -> ps )
	confun4.3	|- ( ps -> ( ph -> ch ) )
	confun4.4	|- ( ( ch -> th ) -> ( ( ph -> th ) <-> ps ) )
	confun4.5	|- ( ta <-> ( ch -> th ) )
	confun4.6	|- ( et <-> -. ( ch -> ( ch /\ -. ch ) ) )
	confun4.7	|- ps
	confun4.8	|- ( ch -> th )
Assertion	confun4	|- ( ch -> ( ps -> ta ) )

Proof

Step	Hyp	Ref	Expression
1		confun4.1	|- ph
2		confun4.2	|- ( ( ph -> ps ) -> ps )
3		confun4.3	|- ( ps -> ( ph -> ch ) )
4		confun4.4	|- ( ( ch -> th ) -> ( ( ph -> th ) <-> ps ) )
5		confun4.5	|- ( ta <-> ( ch -> th ) )
6		confun4.6	|- ( et <-> -. ( ch -> ( ch /\ -. ch ) ) )
7		confun4.7	|- ps
8		confun4.8	|- ( ch -> th )
9	7 3	ax-mp	|- ( ph -> ch )
10	1 9	ax-mp	|- ch
11		bicom1	|- ( ( ta <-> ( ch -> th ) ) -> ( ( ch -> th ) <-> ta ) )
12	5 11	ax-mp	|- ( ( ch -> th ) <-> ta )
13	12	biimpi	|- ( ( ch -> th ) -> ta )
14	8 13	ax-mp	|- ta
15	7 14	pm3.2i	|- ( ps /\ ta )
16		pm3.4	|- ( ( ps /\ ta ) -> ( ps -> ta ) )
17	15 16	ax-mp	|- ( ps -> ta )
18	10 17	pm3.2i	|- ( ch /\ ( ps -> ta ) )
19		pm3.4	|- ( ( ch /\ ( ps -> ta ) ) -> ( ch -> ( ps -> ta ) ) )
20	18 19	ax-mp	|- ( ch -> ( ps -> ta ) )