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Theorem ineq12 3199
Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
Assertion
Ref Expression
ineq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem ineq12
StepHypRef Expression
1 ineq1 3197 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 ineq2 3198 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2141 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  cin 3001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-in 3008
This theorem is referenced by:  ineq12i  3202  ineq12d  3205  ineqan12d  3206  fnun  5135  endisj  6596  sbthlemi8  6729  pm54.43  6881  epttop  11853  bj-inex  12102
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