Theorem reseq2d 5038

Description: Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

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

Assertion

Ref	Expression
reseq2d	|- ( ph -> ( C |` A ) = ( C |` B ) )

Proof of Theorem reseq2d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq2 5033	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1398   |` cres 4751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-opab 4172  df-xp 4755  df-res 4761

This theorem is referenced by:  reseq12d  5039  resima2  5072  relresfld  5292  f1orescnv  5630  funcocnv2  5639  fococnv2  5640  fnressn  5870  oprssov  6196  dftpos2  6492  fnsnsplitdc  6738  dif1en  7136  sbthlemi4  7230  fseq1p1m1  10428  resunimafz0  11198  setsvala  13243  gsumsplit0  14063  metreslem  15245  xmspropd  15342  mspropd  15343  egrsubgr  16258  eupthvdres  16470  eupth2lem3fi  16471  eupth2fi  16474  bj-charfundcALT  16579