MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  necon4abid Structured version   Unicode version

Theorem necon4abid 2682
Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon4abid.1  |-  ( ph  ->  ( A  =/=  B  <->  -. 
ps ) )
Assertion
Ref Expression
necon4abid  |-  ( ph  ->  ( A  =  B  <->  ps ) )

Proof of Theorem necon4abid
StepHypRef Expression
1 notnot 292 . 2  |-  ( ps  <->  -. 
-.  ps )
2 necon4abid.1 . . 3  |-  ( ph  ->  ( A  =/=  B  <->  -. 
ps ) )
32necon1bbid 2681 . 2  |-  ( ph  ->  ( -.  -.  ps  <->  A  =  B ) )
41, 3syl5rbb 261 1  |-  ( ph  ->  ( A  =  B  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    = wceq 1437    =/= wne 2625
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 188  df-ne 2627
This theorem is referenced by:  necon4bbid  2684  necon2bbid  2687  birthdaylem3  23744  nmounbi  26262
  Copyright terms: Public domain W3C validator