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Theorem ineq2d 3282
 Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
ineq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ineq2d (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem ineq2d
StepHypRef Expression
1 ineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ineq2 3276 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∩ cin 3075 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082 This theorem is referenced by:  disjpr2  3595  rint0  3818  riin0  3892  disji2  3930  xpriindim  4685  riinint  4808  reseq2  4822  csbresg  4830  resindm  4869  isoselem  5729  zfz1isolem1  10616  fsumm1  11218  ennnfonelemhf1o  11963  restval  12166  basis1  12254  baspartn  12257  eltg  12261  tgdom  12281  ntrval  12319  resttopon2  12387  restopnb  12390  qtopbasss  12730
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