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Theorem ineq1 3270
Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
Assertion
Ref Expression
ineq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem ineq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2203 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 460 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
3 elin 3259 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elin 3259 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43bitr4g 222 . 2 (𝐴 = 𝐵 → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
65eqrdv 2137 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  cin 3070
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077
This theorem is referenced by:  ineq2  3271  ineq12  3272  ineq1i  3273  ineq1d  3276  dfrab3ss  3354  intprg  3804  inex1g  4064  reseq1  4813  fiintim  6817  uzin2  10773  elrestr  12144  inopn  12186  isbasisg  12227  basis1  12230  basis2  12231  tgval  12234  ntrfval  12285  tgrest  12354  restco  12359  restsn  12365  restopnb  12366  txrest  12461  metrest  12691  qtopbasss  12706  bdinex1g  13206
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