MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon2abii Structured version   Unicode version

Theorem necon2abii 2664
Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
Hypothesis
Ref Expression
necon2abii.1  |-  ( A  =  B  <->  -.  ph )
Assertion
Ref Expression
necon2abii  |-  ( ph  <->  A  =/=  B )

Proof of Theorem necon2abii
StepHypRef Expression
1 necon2abii.1 . . . 4  |-  ( A  =  B  <->  -.  ph )
21bicomi 202 . . 3  |-  ( -. 
ph 
<->  A  =  B )
32necon1abii 2660 . 2  |-  ( A  =/=  B  <->  ph )
43bicomi 202 1  |-  ( ph  <->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1364    =/= wne 2604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-ne 2606
This theorem is referenced by:  flimsncls  19518  tsmsgsum  19668  tsmsgsumOLD  19671  wilthlem2  22366  ismblfin  28357  locfindis  28502  elnev  29617
  Copyright terms: Public domain W3C validator