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Theorem ineq2i 3274
 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 3271 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2ax-mp 5 1 (𝐶𝐴) = (𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∩ 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:  in4  3292  inindir  3294  indif2  3320  difun1  3336  dfrab3ss  3354  dfif3  3487  intunsn  3809  rint0  3810  riin0  3884  res0  4823  resres  4831  resundi  4832  resindi  4834  inres  4836  resiun2  4839  resopab  4863  dfse2  4912  dminxp  4983  imainrect  4984  resdmres  5030  funimacnv  5199  unfiin  6814  sbthlemi5  6849  dmaddpi  7133  dmmulpi  7134  fsumiun  11246
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