Theorem reseq12d 5059

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 5057	. 2 |- ( ph -> ( A |` C ) = ( B |` C ) )
3		reseqd.2	. . 3 |- ( ph -> C = D )
4	3	reseq2d 5058	. 2 |- ( ph -> ( B |` C ) = ( B |` D ) )
5	2, 4	eqtrd 2271	1 |- ( ph -> ( A |` C ) = ( B |` D ) )

Syntax hints:   -> wi 4   = wceq 1402   |` cres 4771

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 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 theorem 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 4188  df-xp 4775  df-res 4781

This theorem is referenced by:  tfrlem3ag  6570  tfrlem3a  6571  tfrlemi1  6593  tfr1onlem3ag  6598  setsvalg  13360  znval  14943  psrval  14973  isxms  15475  isms  15477  issubgr  16412