ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reseq1d GIF version

Theorem reseq1d 4908
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypothesis
Ref Expression
reseqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
reseq1d (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem reseq1d
StepHypRef Expression
1 reseqd.1 . 2 (𝜑𝐴 = 𝐵)
2 reseq1 4903 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  cres 4630
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-res 4640
This theorem is referenced by:  reseq12d  4910  fun2ssres  5261  funcnvres2  5293  funimaexg  5302  fresin  5396  offres  6139  tfrlemisucaccv  6329  tfrlemi1  6336  tfr1onlemsucaccv  6345  tfrcllemsucaccv  6358  freceq1  6396  freceq2  6397  fseq1p1m1  10097  setsresg  12503  setscom  12505  dvcoapbr  14311  bj-charfundcALT  14701
  Copyright terms: Public domain W3C validator