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Theorem reseq2d 4713
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 4708 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cres 4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-opab 3900  df-xp 4444  df-res 4450
This theorem is referenced by:  reseq12d  4714  resima2  4746  relresfld  4960  f1orescnv  5269  funcocnv2  5278  fococnv2  5279  fnressn  5483  oprssov  5786  dftpos2  6026  dif1en  6593  sbthlemi4  6667  fseq1p1m1  9504  resunimafz0  10232  setsvala  11520
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