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Theorem ineq2 3241
Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Assertion
Ref Expression
ineq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem ineq2
StepHypRef Expression
1 ineq1 3240 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 incom 3238 . 2 (𝐶𝐴) = (𝐴𝐶)
3 incom 3238 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2175 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  cin 3040
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047
This theorem is referenced by:  ineq12  3242  ineq2i  3244  ineq2d  3247  uneqin  3297  intprg  3774  fiintim  6785  uzin2  10727  inopn  12097  basis1  12141  basis2  12142  baspartn  12144  metreslem  12476  qtopbasss  12617
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