ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neeq1 Unicode version

Theorem neeq1 2218
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq1  |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )

Proof of Theorem neeq1
StepHypRef Expression
1 eqeq1 2046 . . 3  |-  ( A  =  B  ->  ( A  =  C  <->  B  =  C ) )
21notbid 592 . 2  |-  ( A  =  B  ->  ( -.  A  =  C  <->  -.  B  =  C ) )
3 df-ne 2206 . 2  |-  ( A  =/=  C  <->  -.  A  =  C )
4 df-ne 2206 . 2  |-  ( B  =/=  C  <->  -.  B  =  C )
52, 3, 43bitr4g 212 1  |-  ( A  =  B  ->  ( A  =/=  C  <->  B  =/=  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 98    = wceq 1243    =/= wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-ne 2206
This theorem is referenced by:  neeq1i  2220  neeq1d  2223  nelrdva  2746  psseq1  3031  0inp0  3919  uzn0  8488  xrnemnf  8699  xrnepnf  8700  ngtmnft  8731  fztpval  8945
  Copyright terms: Public domain W3C validator