Theorem reseq2d 4958

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 4953	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1372   |` cres 4676

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-opab 4105  df-xp 4680  df-res 4686

This theorem is referenced by:  reseq12d  4959  resima2  4992  relresfld  5211  f1orescnv  5537  funcocnv2  5546  fococnv2  5547  fnressn  5769  oprssov  6087  dftpos2  6346  fnsnsplitdc  6590  dif1en  6975  sbthlemi4  7061  fseq1p1m1  10215  resunimafz0  10974  setsvala  12834  metreslem  14823  xmspropd  14920  mspropd  14921  bj-charfundcALT  15707