Theorem reseq2d 5063

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 5058	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   |` cres 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-opab 4193  df-xp 4780  df-res 4786

This theorem is used by:  reseq12d  5064  resima2  5097  relresfld  5317  f1orescnv  5655  funcocnv2  5664  fococnv2  5665  fnressn  5901  oprssov  6231  dftpos2  6532  fnsnsplitdc  6778  dif1en  7183  sbthlemi4  7277  fseq1p1m1  10501  resunimafz0  11274  setsvala  13383  gzsumsplit0  14148  metreslem  15481  xmspropd  15578  mspropd  15579  egrsubgr  16504  eupthvdres  16716  eupth2lem3fi  16717  eupth2fi  16720  bj-charfundcALT  16835