Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  reseq12i Structured version   Visualization version   GIF version

Theorem reseq12i 5532
 Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
reseqi.1 𝐴 = 𝐵
reseqi.2 𝐶 = 𝐷
Assertion
Ref Expression
reseq12i (𝐴𝐶) = (𝐵𝐷)

Proof of Theorem reseq12i
StepHypRef Expression
1 reseqi.1 . . 3 𝐴 = 𝐵
21reseq1i 5530 . 2 (𝐴𝐶) = (𝐵𝐶)
3 reseqi.2 . . 3 𝐶 = 𝐷
43reseq2i 5531 . 2 (𝐵𝐶) = (𝐵𝐷)
52, 4eqtri 2793 1 (𝐴𝐶) = (𝐵𝐷)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ↾ cres 5251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-opab 4847  df-xp 5255  df-res 5261 This theorem is referenced by:  cnvresid  6108  wfrlem5  7572  dfoi  8572  lubfval  17186  glbfval  17199  oduglb  17347  odulub  17349  dvlog  24618  dvlog2  24620  issubgr  26386  finsumvtxdg2size  26681  sitgclg  30744  frrlem5  32121  fourierdlem57  40897  fourierdlem74  40914  fourierdlem75  40915
 Copyright terms: Public domain W3C validator