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Theorem eqnetri 2305
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
eqnetr.1  |-  A  =  B
eqnetr.2  |-  B  =/= 
C
Assertion
Ref Expression
eqnetri  |-  A  =/= 
C

Proof of Theorem eqnetri
StepHypRef Expression
1 eqnetr.2 . 2  |-  B  =/= 
C
2 eqnetr.1 . . 3  |-  A  =  B
32neeq1i 2297 . 2  |-  ( A  =/=  C  <->  B  =/=  C )
41, 3mpbir 145 1  |-  A  =/= 
C
Colors of variables: wff set class
Syntax hints:    = wceq 1314    =/= wne 2282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-ne 2283
This theorem is referenced by:  eqnetrri  2307  2on0  6277  1n0  6283  basendxnplusgndx  11905  plusgndxnmulrndx  11912  basendxnmulrndx  11913
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