Metamath Proof Explorer

Theorem necon4bbid

Description: Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012)

Ref Expression
Hypothesis necon4bbid.1 
|- ( ph -> ( -. ps <-> A =/= B ) )
Assertion necon4bbid 
|- ( ph -> ( ps <-> A = B ) )

Proof

Step Hyp Ref Expression
1 necon4bbid.1 
 |-  ( ph -> ( -. ps <-> A =/= B ) )
2 1 bicomd 
 |-  ( ph -> ( A =/= B <-> -. ps ) )
3 2 necon4abid 
 |-  ( ph -> ( A = B <-> ps ) )
4 3 bicomd 
 |-  ( ph -> ( ps <-> A = B ) )