MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon1abii Structured version   Unicode version

Theorem necon1abii 2649
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
Hypothesis
Ref Expression
necon1abii.1  |-  ( -. 
ph 
<->  A  =  B )
Assertion
Ref Expression
necon1abii  |-  ( A  =/=  B  <->  ph )

Proof of Theorem necon1abii
StepHypRef Expression
1 df-ne 2600 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1abii.1 . . 3  |-  ( -. 
ph 
<->  A  =  B )
32con1bii 322 . 2  |-  ( -.  A  =  B  <->  ph )
41, 3bitri 241 1  |-  ( A  =/=  B  <->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    =/= wne 2598
This theorem is referenced by:  necon2abii  2653  marypha1lem  7430  npomex  8865  restutopopn  18260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 178  df-ne 2600
  Copyright terms: Public domain W3C validator