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

Theorem inteqd 3847
Description: Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
Hypothesis
Ref Expression
inteqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
inteqd  |-  ( ph  ->  |^| A  =  |^| B )

Proof of Theorem inteqd
StepHypRef Expression
1 inteqd.1 . 2  |-  ( ph  ->  A  =  B )
2 inteq 3845 . 2  |-  ( A  =  B  ->  |^| A  =  |^| B )
31, 2syl 14 1  |-  ( ph  ->  |^| A  =  |^| B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   |^|cint 3842
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-ral 2460  df-int 3843
This theorem is referenced by:  intprg  3875  op1stbg  4475  onsucmin  4502  elreldm  4848  elxp5  5112  fniinfv  5569  1stval2  6149  2ndval2  6150  fundmen  6799  xpsnen  6814  fiintim  6921  elfi2  6964  fi0  6967  cardcl  7173  isnumi  7174  cardval3ex  7177  carden2bex  7181  clsfval  13234  clsval  13244
  Copyright terms: Public domain W3C validator