Theorem reseq1d 4629

Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
reseq1d	|- ( ph -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq1 4624	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1284   |` cres 4365

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-res 4375

This theorem is referenced by:  reseq12d  4631  fun2ssres  4963  funcnvres2  4994  funimaexg  5003  fresin  5088  offres  5782  tfrlemisucaccv  5962  tfrlemi1  5969  freceq1  6002  freceq2  6003  fseq1p1m1  9111