Theorem bj-andnotim 32573

Description: Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-andnotim	|- ( ( ( ph /\ -. ps ) -> ch ) <-> ( ( ph -> ps ) \/ ch ) )

Proof of Theorem bj-andnotim

Step	Hyp	Ref	Expression
1		imor 428	. . 3 |- ( ( ( ph /\ -. ps ) -> ch ) <-> ( -. ( ph /\ -. ps ) \/ ch ) )
2		iman 440	. . . . 5 |- ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) )
3	2	biimpri 218	. . . 4 |- ( -. ( ph /\ -. ps ) -> ( ph -> ps ) )
4	3	orim1i 539	. . 3 |- ( ( -. ( ph /\ -. ps ) \/ ch ) -> ( ( ph -> ps ) \/ ch ) )
5	1, 4	sylbi 207	. 2 |- ( ( ( ph /\ -. ps ) -> ch ) -> ( ( ph -> ps ) \/ ch ) )
6		pm2.24 121	. . . . 5 |- ( ps -> ( -. ps -> ch ) )
7	6	imim2i 16	. . . 4 |- ( ( ph -> ps ) -> ( ph -> ( -. ps -> ch ) ) )
8	7	impd 447	. . 3 |- ( ( ph -> ps ) -> ( ( ph /\ -. ps ) -> ch ) )
9		ax-1 6	. . 3 |- ( ch -> ( ( ph /\ -. ps ) -> ch ) )
10	8, 9	jaoi 394	. 2 |- ( ( ( ph -> ps ) \/ ch ) -> ( ( ph /\ -. ps ) -> ch ) )
11	5, 10	impbii 199	1 |- ( ( ( ph /\ -. ps ) -> ch ) <-> ( ( ph -> ps ) \/ ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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-or 385  df-an 386

This theorem is referenced by: (None)