Theorem reseq12d 5986

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 5984	. 2 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3		reseqd.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	reseq2d 5985	. 2 ⊢ (𝜑 → (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷))
5	2, 4	eqtrd 2765	1 ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷))

Syntax hints:   → wi 4   = wceq 1533   ↾ cres 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-in 3951  df-opab 5212  df-xp 5684  df-res 5690

This theorem is referenced by:  f1ossf1o  7137  csbfrecsg  8290  tfrlem3a  8398  oieq1  9537  oieq2  9538  ackbij2lem3  10266  setsvalg  17138  resfval  17881  resfval2  17882  resf2nd  17884  lubfval  18345  glbfval  18358  dpjfval  20024  znval  21482  psrval  21865  prdsdsf  24317  prdsxmet  24319  imasdsf1olem  24323  xpsxmetlem  24329  xpsmet  24332  isxms  24397  isms  24399  setsxms  24431  setsms  24432  ressxms  24478  ressms  24479  prdsxmslem2  24482  cphsscph  25223  iscms  25317  cmsss  25323  cssbn  25347  minveclem3a  25399  dvmptresicc  25889  dvcmulf  25920  efcvx  26431  issubgr  29156  ispth  29609  clwlknf1oclwwlkn  29966  eucrct2eupth  30127  padct  32583  isrrext  33732  prdsbnd2  37399  cnpwstotbnd  37401  ldualset  38727  itgcoscmulx  45495  fourierdlem73  45705  sge0fodjrnlem  45942  vonval  46066  dfateq12d  46644  isisubgr  47334  rngchomrnghmresALTV  47527  fdivval  47798