Theorem reseq12d 4957

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 4955	. 2 |- ( ph -> ( A |` C ) = ( B |` C ) )
3		reseqd.2	. . 3 |- ( ph -> C = D )
4	3	reseq2d 4956	. 2 |- ( ph -> ( B |` C ) = ( B |` D ) )
5	2, 4	eqtrd 2237	1 |- ( ph -> ( A |` C ) = ( B |` D ) )

Syntax hints:   -> wi 4   = wceq 1372   |` cres 4675

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-opab 4105  df-xp 4679  df-res 4685

This theorem is referenced by:  tfrlem3ag  6385  tfrlem3a  6386  tfrlemi1  6408  tfr1onlem3ag  6413  setsvalg  12781  znval  14316  psrval  14346  isxms  14841  isms  14843