Theorem 3impexp 1289

Description: Version of impexp 462 for a triple conjunction. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
3impexp	|- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Proof of Theorem 3impexp

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ( ph /\ ps /\ ch ) -> th ) )
2	1	3expd 1284	. 2 |- ( ( ( ph /\ ps /\ ch ) -> th ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
3		id 22	. . 3 |- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ps -> ( ch -> th ) ) ) )
4	3	3impd 1281	. 2 |- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ( ph /\ ps /\ ch ) -> th ) )
5	2, 4	impbii 199	1 |- ( ( ( ph /\ ps /\ ch ) -> th ) <-> ( ph -> ( ps -> ( ch -> th ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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:  cotr2g  13715  bnj978  31019  3impexpbicom  38685  3impexpbicomVD  39092