Theorem reseq12d 4867

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 4865	. 2 |- ( ph -> ( A |` C ) = ( B |` C ) )
3		reseqd.2	. . 3 |- ( ph -> C = D )
4	3	reseq2d 4866	. 2 |- ( ph -> ( B |` C ) = ( B |` D ) )
5	2, 4	eqtrd 2190	1 |- ( ph -> ( A |` C ) = ( B |` D ) )

Syntax hints:   -> wi 4   = wceq 1335   |` cres 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-opab 4026  df-xp 4592  df-res 4598

This theorem is referenced by:  tfrlem3ag  6256  tfrlem3a  6257  tfrlemi1  6279  tfr1onlem3ag  6284  setsvalg  12231  isxms  12862  isms  12864