MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon1bbii Unicode version

Theorem necon1bbii 2602
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
Hypothesis
Ref Expression
necon1bbii.1  |-  ( A  =/=  B  <->  ph )
Assertion
Ref Expression
necon1bbii  |-  ( -. 
ph 
<->  A  =  B )

Proof of Theorem necon1bbii
StepHypRef Expression
1 df-ne 2552 . . 3  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1bbii.1 . . 3  |-  ( A  =/=  B  <->  ph )
31, 2bitr3i 243 . 2  |-  ( -.  A  =  B  <->  ph )
43con1bii 322 1  |-  ( -. 
ph 
<->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    =/= wne 2550
This theorem is referenced by:  necon2bbii  2606  rabeq0  3592  intnex  4298  class2set  4308  relimasn  5167  modom  7245  fzo0  11089  vma1  20816  lgsquadlem3  21007
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 2552
  Copyright terms: Public domain W3C validator