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Theorem eqnetrd 2371
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 2365 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
41, 3mpbird 167 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wne 2347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-ne 2348
This theorem is referenced by:  eqnetrrd  2373  frecabcl  6397  frecsuclem  6404  omp1eomlem  7090  xaddnemnf  9853  xaddnepnf  9854  hashprg  10781  bezoutr1  12026  phibndlem  12208  dfphi2  12212  lgsne0  14310  2sqlem8a  14329  2sqlem8  14330
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