Theorem reseq2d 4973

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 4968	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1373   |` cres 4690

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-opab 4117  df-xp 4694  df-res 4700

This theorem is referenced by:  reseq12d  4974  resima2  5007  relresfld  5226  f1orescnv  5555  funcocnv2  5564  fococnv2  5565  fnressn  5788  oprssov  6106  dftpos2  6365  fnsnsplitdc  6609  dif1en  6997  sbthlemi4  7083  fseq1p1m1  10246  resunimafz0  11008  setsvala  12948  metreslem  14937  xmspropd  15034  mspropd  15035  bj-charfundcALT  15914