Theorem notbi 309

Description: Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)

Assertion

Ref	Expression
notbi	|- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )

Proof of Theorem notbi

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( ( ph <-> ps ) -> ( ph <-> ps ) )
2	1	notbid 308	. 2 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )
3		id 22	. . 3 |- ( ( -. ph <-> -. ps ) -> ( -. ph <-> -. ps ) )
4	3	con4bid 307	. 2 |- ( ( -. ph <-> -. ps ) -> ( ph <-> ps ) )
5	2, 4	impbii 199	1 |- ( ( ph <-> ps ) <-> ( -. ph <-> -. ps ) )

Syntax hints:   -. wn 3   <-> 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:  notbii  310  con4bii  311  con2bi  343  nbn2  360  pm5.32  668  hadnot  1541  had0  1543  cbvexd  2278  symdifass  3853  isocnv3  6582  suppimacnv  7306  sumodd  15111  f1omvdco3  17869  onsuct0  32440  bj-cbvexdv  32736  ifpbi1  37822  ifpbi13  37834  abciffcbatnabciffncba  41096  abciffcbatnabciffncbai  41097