Theorem confun4 41109

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

Hypotheses

Ref	Expression
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

Ref	Expression
confun4	|- ( ch -> ( ps -> ta ) )

Proof of Theorem confun4

Step	Hyp	Ref	Expression
1		confun4.1	. . . 4 |- ph
2		confun4.7	. . . . 5 |- ps
3		confun4.3	. . . . 5 |- ( ps -> ( ph -> ch ) )
4	2, 3	ax-mp 5	. . . 4 |- ( ph -> ch )
5	1, 4	ax-mp 5	. . 3 |- ch
6		confun4.8	. . . . . 6 |- ( ch -> th )
7		confun4.5	. . . . . . . 8 |- ( ta <-> ( ch -> th ) )
8		bicom1 211	. . . . . . . 8 |- ( ( ta <-> ( ch -> th ) ) -> ( ( ch -> th ) <-> ta ) )
9	7, 8	ax-mp 5	. . . . . . 7 |- ( ( ch -> th ) <-> ta )
10	9	biimpi 206	. . . . . 6 |- ( ( ch -> th ) -> ta )
11	6, 10	ax-mp 5	. . . . 5 |- ta
12	2, 11	pm3.2i 471	. . . 4 |- ( ps /\ ta )
13		pm3.4 584	. . . 4 |- ( ( ps /\ ta ) -> ( ps -> ta ) )
14	12, 13	ax-mp 5	. . 3 |- ( ps -> ta )
15	5, 14	pm3.2i 471	. 2 |- ( ch /\ ( ps -> ta ) )
16		pm3.4 584	. 2 |- ( ( ch /\ ( ps -> ta ) ) -> ( ch -> ( ps -> ta ) ) )
17	15, 16	ax-mp 5	1 |- ( ch -> ( ps -> 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

This theorem is referenced by: (None)