This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem necon2ai

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 22-Nov-2019)

Ref Expression
Hypothesis necon2ai.1 
|- ( A = B -> -. ph )
Assertion necon2ai 
|- ( ph -> A =/= B )

Proof

Step Hyp Ref Expression
1 necon2ai.1 
 |-  ( A = B -> -. ph )
2 1 con2i 
 |-  ( ph -> -. A = B )
3 2 neqned 
 |-  ( ph -> A =/= B )