Theorem ineq1i 3810

Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Hypothesis

Ref	Expression
ineq1i.1	|- A = B

Assertion

Ref	Expression
ineq1i	|- ( A i^i C ) = ( B i^i C )

Proof of Theorem ineq1i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 |- A = B
2		ineq1 3807	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
3	1, 2	ax-mp 5	1 |- ( A i^i C ) = ( B i^i C )

Syntax hints:   = wceq 1483   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581

This theorem is referenced by:  in12  3824  inindi  3830  dfrab3  3902  dfif5  4102  disjpr2  4248  disjpr2OLD  4249  disjtpsn  4251  disjtp2  4252  resres  5409  imainrect  5575  predidm  5702  fresaun  6075  fresaunres2  6076  ssenen  8134  hartogslem1  8447  prinfzo0  12506  leiso  13243  f1oun2prg  13662  smumul  15215  setsfun  15893  setsfun0  15894  firest  16093  lsmdisj2r  18098  frgpuplem  18185  ltbwe  19472  tgrest  20963  fiuncmp  21207  ptclsg  21418  metnrmlem3  22664  mbfid  23403  ppi1  24890  cht1  24891  ppiub  24929  chdmj2i  28341  chjassi  28345  pjoml2i  28444  pjoml4i  28446  cmcmlem  28450  mayetes3i  28588  cvmdi  29183  atomli  29241  atabsi  29260  uniin1  29367  disjuniel  29410  imadifxp  29414  gtiso  29478  prsss  29962  ordtrest2NEW  29969  esumnul  30110  measinblem  30283  eulerpartlemt  30433  ballotlem2  30550  ballotlemfp1  30553  ballotlemfval0  30557  chtvalz  30707  mthmpps  31479  dffv5  32031  bj-sscon  33014  bj-discrmoore  33066  mblfinlem2  33447  ismblfin  33450  mbfposadd  33457  itg2addnclem2  33462  asindmre  33495  abeqin  34017  diophrw  37322  dnwech  37618  lmhmlnmsplit  37657  rp-fakeuninass  37862  iunrelexp0  37994  nznngen  38515  uzinico2  39789  limsup0  39926  limsupvaluz  39940  sge0sn  40596  31prm  41512