Theorem reseq12d 5039

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

Hypotheses

Ref	Expression
reseqd.1	|- ( ph -> A = B )
reseqd.2	|- ( ph -> C = D )

Assertion

Ref	Expression
reseq12d	|- ( ph -> ( A |` C ) = ( B |` D ) )

Proof of Theorem reseq12d

Step	Hyp	Ref	Expression
1		reseqd.1	. . 3 |- ( ph -> A = B )
2	1	reseq1d 5037	. 2 |- ( ph -> ( A |` C ) = ( B |` C ) )
3		reseqd.2	. . 3 |- ( ph -> C = D )
4	3	reseq2d 5038	. 2 |- ( ph -> ( B |` C ) = ( B |` D ) )
5	2, 4	eqtrd 2265	1 |- ( ph -> ( A |` C ) = ( B |` D ) )

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:  tfrlem3ag  6540  tfrlem3a  6541  tfrlemi1  6563  tfr1onlem3ag  6568  setsvalg  13242  znval  14784  psrval  14814  isxms  15316  isms  15318  issubgr  16252