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Theorem ineq12d 3694
Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
ineq1d.1  |-  ( ph  ->  A  =  B )
ineq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
ineq12d  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )

Proof of Theorem ineq12d
StepHypRef Expression
1 ineq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ineq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 ineq12 3688 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3syl2anc 661 1  |-  ( ph  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    i^i cin 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-in 3476
This theorem is referenced by:  csbin  3853  csbingOLD  3854  funprg  5628  funtpg  5629  offval  6522  ofrfval  6523  oev2  7163  isf32lem7  8728  ressval  14531  invffval  15002  invfval  15003  oppcinv  15020  isps  15678  dmdprd  16813  dprddisj  16826  dprdf1o  16862  dmdprdsplit2lem  16877  dmdprdpr  16881  pgpfaclem1  16915  isunit  17083  dfrhm2  17143  isrhm  17147  2idlval  17656  aspval  17741  ressmplbas2  17881  pjfval  18497  iscon  19673  consuba  19680  ptbasin  19806  ptclsg  19844  qtopval  19924  rnelfmlem  20181  trust  20460  isnmhm  20981  uniioombllem2a  21719  dyaddisjlem  21732  dyaddisj  21733  i1faddlem  21828  i1fmullem  21829  limcflf  22013  ispth  24232  1pthonlem2  24254  2pthlem2  24260  constr2pth  24265  constr3pthlem3  24319  frisusgranb  24659  2spotdisj  24724  chocin  26075  cmbr3  26188  pjoml3  26192  fh1  26198  xppreima2  27146  hauseqcn  27499  prsssdm  27521  ordtrestNEW  27525  ordtrest2NEW  27527  cndprobval  27998  ballotlemfrc  28091  predeq123  28808  itg2addnclem2  29631  clsun  29710  heiborlem4  29900  heiborlem6  29902  heiborlem10  29906  aomclem8  30600  dvsinax  31196  dvcosax  31211  usgra2pthspth  31775  bnj1421  33052  pmodl42N  34522  polfvalN  34575  poldmj1N  34599  pmapj2N  34600  pnonsingN  34604  psubclinN  34619  poml4N  34624  osumcllem9N  34635  trnfsetN  34826  diainN  35729  djaffvalN  35805  djafvalN  35806  djajN  35809  dihmeetcl  36017  dihmeet2  36018  dochnoncon  36063  djhffval  36068  djhfval  36069  djhlj  36073  dochdmm1  36082  lclkrlem2g  36185  lclkrlem2v  36200  lcfrlem21  36235  lcfrlem24  36238  mapdunirnN  36322  baerlem5amN  36388  baerlem5bmN  36389  baerlem5abmN  36390  mapdheq4lem  36403  mapdh6lem1N  36405  mapdh6lem2N  36406  hdmap1l6lem1  36480  hdmap1l6lem2  36481
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