Theorem df3or2 38060

Description: Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)

Assertion

Ref	Expression
df3or2	|- ( ( ph \/ ps \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )

Proof of Theorem df3or2

Step	Hyp	Ref	Expression
1		df-3or 1038	. 2 |- ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
2		df-or 385	. . 3 |- ( ( ( ph \/ ps ) \/ ch ) <-> ( -. ( ph \/ ps ) -> ch ) )
3		ioran 511	. . . . 5 |- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) )
4	3	imbi1i 339	. . . 4 |- ( ( -. ( ph \/ ps ) -> ch ) <-> ( ( -. ph /\ -. ps ) -> ch ) )
5		impexp 462	. . . 4 |- ( ( ( -. ph /\ -. ps ) -> ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
6	4, 5	bitri 264	. . 3 |- ( ( -. ( ph \/ ps ) -> ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
7	2, 6	bitri 264	. 2 |- ( ( ( ph \/ ps ) \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )
8	1, 7	bitri 264	1 |- ( ( ph \/ ps \/ ch ) <-> ( -. ph -> ( -. ps -> ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036

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  df-3or 1038

This theorem is referenced by: (None)