Theorem impbi 198

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)

Assertion

Ref	Expression
impbi	|- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) )

Proof of Theorem impbi

Step	Hyp	Ref	Expression
1		df-bi 197	. . 3 |- -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) )
2		simprim 162	. . 3 |- ( -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) ) -> ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) )
3	1, 2	ax-mp 5	. 2 |- ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) )
4	3	expi 161	1 |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196

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

This theorem is referenced by:  impbii  199  impbidd  200  dfbi1  203  bj-bisym  32575  eqsbc3rVD  39075  orbi1rVD  39083  3impexpVD  39091  3impexpbicomVD  39092  imbi12VD  39109  sbcim2gVD  39111  sb5ALTVD  39149