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Theorem reseq12d 4714
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 4712 . 2  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
3 reseqd.2 . . 3  |-  ( ph  ->  C  =  D )
43reseq2d 4713 . 2  |-  ( ph  ->  ( B  |`  C )  =  ( B  |`  D ) )
52, 4eqtrd 2120 1  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
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:  tfrlem3ag  6074  tfrlem3a  6075  tfrlemi1  6097  tfr1onlem3ag  6102  setsvalg  11519
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