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Theorem eqnetri 2308
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
eqnetr.1 𝐴 = 𝐵
eqnetr.2 𝐵𝐶
Assertion
Ref Expression
eqnetri 𝐴𝐶

Proof of Theorem eqnetri
StepHypRef Expression
1 eqnetr.2 . 2 𝐵𝐶
2 eqnetr.1 . . 3 𝐴 = 𝐵
32neeq1i 2300 . 2 (𝐴𝐶𝐵𝐶)
41, 3mpbir 145 1 𝐴𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-ne 2286
This theorem is referenced by:  eqnetrri  2310  2on0  6291  1n0  6297  basendxnplusgndx  11992  plusgndxnmulrndx  11999  basendxnmulrndx  12000
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