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Theorem reseq12d 4892
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 4890 . 2  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
3 reseqd.2 . . 3  |-  ( ph  ->  C  =  D )
43reseq2d 4891 . 2  |-  ( ph  ->  ( B  |`  C )  =  ( B  |`  D ) )
52, 4eqtrd 2203 1  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    |` cres 4613
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-opab 4051  df-xp 4617  df-res 4623
This theorem is referenced by:  tfrlem3ag  6288  tfrlem3a  6289  tfrlemi1  6311  tfr1onlem3ag  6316  setsvalg  12446  isxms  13245  isms  13247
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