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Theorem reseq12d 5106
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
reseqd.1  |-  ( ph  ->  A  =  B )
reseqd.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
reseq12d  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )

Proof of Theorem reseq12d
StepHypRef Expression
1 reseqd.1 . . 3  |-  ( ph  ->  A  =  B )
21reseq1d 5104 . 2  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
3 reseqd.2 . . 3  |-  ( ph  ->  C  =  D )
43reseq2d 5105 . 2  |-  ( ph  ->  ( B  |`  C )  =  ( B  |`  D ) )
52, 4eqtrd 2470 1  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    |` cres 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969  df-in 3330  df-opab 4346  df-xp 4841  df-res 4847
This theorem is referenced by:  tfrlem3a  6828  oieq1  7718  oieq2  7719  ackbij2lem3  8402  setsvalg  14189  resfval  14794  resfval2  14795  resf2nd  14797  lubfval  15140  glbfval  15153  dpjfval  16542  psrval  17406  znval  17941  prdsdsf  19917  prdsxmet  19919  imasdsf1olem  19923  xpsxmetlem  19929  xpsmet  19932  isxms  19997  isms  19999  setsxms  20029  setsms  20030  ressxms  20075  ressms  20076  prdsxmslem2  20079  iscms  20831  cmsss  20836  minveclem3a  20889  dvcmulf  21394  efcvx  21889  ispth  23418  constr3pthlem1  23492  isrrext  26381  prdsbnd2  28647  cnpwstotbnd  28649  dfateq12d  29988  ldualset  32610
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