Theorem reseq12d 5982

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

Hypotheses

Ref	Expression
reseqd.1	⊢ (𝜑 → 𝐴 = 𝐵)
reseqd.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
reseq12d	⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷))

Proof of Theorem reseq12d

Step	Hyp	Ref	Expression
1		reseqd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	reseq1d 5980	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3		reseqd.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	reseq2d 5981	. 2 ⊢ (𝜑 → (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷))
5	2, 4	eqtrd 2771	1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷))

Syntax hints:   → wi 4   = wceq 1540   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-in 3955  df-opab 5211  df-xp 5682  df-res 5688

This theorem is referenced by:  f1ossf1o  7128  csbfrecsg  8273  tfrlem3a  8381  oieq1  9511  oieq2  9512  ackbij2lem3  10240  setsvalg  17104  resfval  17847  resfval2  17848  resf2nd  17850  lubfval  18308  glbfval  18321  dpjfval  19967  znval  21307  psrval  21688  prdsdsf  24094  prdsxmet  24096  imasdsf1olem  24100  xpsxmetlem  24106  xpsmet  24109  isxms  24174  isms  24176  setsxms  24208  setsms  24209  ressxms  24255  ressms  24256  prdsxmslem2  24259  cphsscph  25000  iscms  25094  cmsss  25100  cssbn  25124  minveclem3a  25176  dvmptresicc  25666  dvcmulf  25697  efcvx  26198  issubgr  28796  ispth  29248  clwlknf1oclwwlkn  29605  eucrct2eupth  29766  padct  32212  isrrext  33279  prdsbnd2  36967  cnpwstotbnd  36969  ldualset  38299  itgcoscmulx  44984  fourierdlem73  45194  sge0fodjrnlem  45431  vonval  45555  dfateq12d  46133  rngchomrnghmresALTV  46983  fdivval  47313