Metamath Proof Explorer

Theorem necon1bbid

Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008)

Ref Expression
Hypothesis necon1bbid.1 
|- ( ph -> ( A =/= B <-> ps ) )
Assertion necon1bbid 
|- ( ph -> ( -. ps <-> A = B ) )

Proof

Step Hyp Ref Expression
1 necon1bbid.1 
 |-  ( ph -> ( A =/= B <-> ps ) )
2 df-ne 
 |-  ( A =/= B <-> -. A = B )
3 2 1 bitr3id 
 |-  ( ph -> ( -. A = B <-> ps ) )
4 3 con1bid 
 |-  ( ph -> ( -. ps <-> A = B ) )