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Theorem ineq2i 3163
Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Hypothesis
Ref Expression
ineq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
ineq2i (𝐶𝐴) = (𝐶𝐵)

Proof of Theorem ineq2i
StepHypRef Expression
1 ineq1i.1 . 2 𝐴 = 𝐵
2 ineq2 3160 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2ax-mp 7 1 (𝐶𝐴) = (𝐶𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952
This theorem is referenced by:  in4  3181  inindir  3183  indif2  3209  difun1  3225  dfrab3ss  3243  dfif3  3371  intunsn  3681  rint0  3682  riin0  3756  res0  4644  resres  4652  resundi  4653  resindi  4655  inres  4657  resiun2  4659  resopab  4680  dfse2  4726  dminxp  4793  imainrect  4794  resdmres  4840  funimacnv  5003  dmaddpi  6481  dmmulpi  6482
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