Theorem df3an2 38061

Description: Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)

Assertion

Ref	Expression
df3an2	|- ( ( ph /\ ps /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )

Proof of Theorem df3an2

Step	Hyp	Ref	Expression
1		df-3an 1039	. 2 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2		df-an 386	. . 3 |- ( ( ( ph /\ ps ) /\ ch ) <-> -. ( ( ph /\ ps ) -> -. ch ) )
3		impexp 462	. . 3 |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ph -> ( ps -> -. ch ) ) )
4	2, 3	xchbinx 324	. 2 |- ( ( ( ph /\ ps ) /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )
5	1, 4	bitri 264	1 |- ( ( ph /\ ps /\ ch ) <-> -. ( ph -> ( ps -> -. ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037

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-3an 1039

This theorem is referenced by: (None)