Metamath Proof Explorer

Theorem necon3bd

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Hypothesis necon3bd.1 
|- ( ph -> ( A = B -> ps ) )
Assertion necon3bd 
|- ( ph -> ( -. ps -> A =/= B ) )

Proof

Step Hyp Ref Expression
1 necon3bd.1 
 |-  ( ph -> ( A = B -> ps ) )
2 nne 
 |-  ( -. A =/= B <-> A = B )
3 2 1 biimtrid 
 |-  ( ph -> ( -. A =/= B -> ps ) )
4 3 con1d 
 |-  ( ph -> ( -. ps -> A =/= B ) )