Metamath Proof Explorer

Theorem necon1abii

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007) (Proof shortened by Wolf Lammen, 25-Nov-2019)

Ref Expression
Hypothesis necon1abii.1 
|- ( -. ph <-> A = B )
Assertion necon1abii 
|- ( A =/= B <-> ph )

Proof

Step Hyp Ref Expression
1 necon1abii.1 
 |-  ( -. ph <-> A = B )
2 notnotb 
 |-  ( ph <-> -. -. ph )
3 1 necon3bbii 
 |-  ( -. -. ph <-> A =/= B )
4 2 3 bitr2i 
 |-  ( A =/= B <-> ph )