Theorem reseq12d 4947

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 4945	. 2 |- ( ph -> ( A |` C ) = ( B |` C ) )
3		reseqd.2	. . 3 |- ( ph -> C = D )
4	3	reseq2d 4946	. 2 |- ( ph -> ( B |` C ) = ( B |` D ) )
5	2, 4	eqtrd 2229	1 |- ( ph -> ( A |` C ) = ( B |` D ) )

Syntax hints:   -> wi 4   = wceq 1364   |` cres 4665

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-opab 4095  df-xp 4669  df-res 4675

This theorem is referenced by:  tfrlem3ag  6367  tfrlem3a  6368  tfrlemi1  6390  tfr1onlem3ag  6395  setsvalg  12708  znval  14192  psrval  14220  isxms  14687  isms  14689