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Theorem reseq2d 4819
Description: Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
reseqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
reseq2d (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem reseq2d
StepHypRef Expression
1 reseqd.1 . 2 (𝜑𝐴 = 𝐵)
2 reseq2 4814 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  cres 4541
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  df-opab 3990  df-xp 4545  df-res 4551
This theorem is referenced by:  reseq12d  4820  resima2  4853  relresfld  5068  f1orescnv  5383  funcocnv2  5392  fococnv2  5393  fnressn  5606  oprssov  5912  dftpos2  6158  fnsnsplitdc  6401  dif1en  6773  sbthlemi4  6848  fseq1p1m1  9881  resunimafz0  10581  setsvala  12000  metreslem  12559  xmspropd  12656  mspropd  12657
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