Metamath Proof Explorer

Theorem con2bi

Description: Contraposition. Theorem *4.12 of WhiteheadRussell p. 117. (Contributed by NM, 15-Apr-1995) (Proof shortened by Wolf Lammen, 3-Jan-2013)

Ref Expression
Assertion con2bi 
|- ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) )

Proof

Step Hyp Ref Expression
1 notbi 
 |-  ( ( ph <-> -. ps ) <-> ( -. ph <-> -. -. ps ) )
2 notnotb 
 |-  ( ps <-> -. -. ps )
3 2 bibi2i 
 |-  ( ( -. ph <-> ps ) <-> ( -. ph <-> -. -. ps ) )
4 bicom 
 |-  ( ( -. ph <-> ps ) <-> ( ps <-> -. ph ) )
5 1 3 4 3bitr2i 
 |-  ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) )