MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neeq2i Unicode version

Theorem neeq2i 2586
Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.)
Hypothesis
Ref Expression
neeq1i.1  |-  A  =  B
Assertion
Ref Expression
neeq2i  |-  ( C  =/=  A  <->  C  =/=  B )

Proof of Theorem neeq2i
StepHypRef Expression
1 neeq1i.1 . 2  |-  A  =  B
2 neeq2 2584 . 2  |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
31, 2ax-mp 8 1  |-  ( C  =/=  A  <->  C  =/=  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    =/= wne 2575
This theorem is referenced by:  neeq12i  2587  neeqtri  2596  divnumden2  24122  nosgnn0  25534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-ex 1548  df-cleq 2405  df-ne 2577
  Copyright terms: Public domain W3C validator