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Theorem eqnetrd 2309
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
eqnetrd.1 (𝜑𝐴 = 𝐵)
eqnetrd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
eqnetrd (𝜑𝐴𝐶)

Proof of Theorem eqnetrd
StepHypRef Expression
1 eqnetrd.2 . 2 (𝜑𝐵𝐶)
2 eqnetrd.1 . . 3 (𝜑𝐴 = 𝐵)
32neeq1d 2303 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 166 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  eqnetrrd  2311  frecabcl  6264  frecsuclem  6271  omp1eomlem  6947  xaddnemnf  9608  xaddnepnf  9609  hashprg  10522  bezoutr1  11648  phibndlem  11819  dfphi2  11823
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