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Theorem ineq1 3128
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 2101 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 438 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑥𝐶) ↔ (𝑥𝐵𝑥𝐶)))
3 elin 3123 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elin 3123 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43bitr4g 212 . 2 (𝐴 = 𝐵 → (𝑥 ∈ (𝐴𝐶) ↔ 𝑥 ∈ (𝐵𝐶)))
65eqrdv 2038 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  cin 2913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921
This theorem is referenced by:  ineq2  3129  ineq12  3130  ineq1i  3131  ineq1d  3134  dfrab3ss  3212  intprg  3645  inex1g  3890  reseq1  4584  uzin2  9455  bdinex1g  9885
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