ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq1i Unicode version

Theorem ineq1i 3273
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
StepHypRef Expression
1 ineq1i.1 . 2  |-  A  =  B
2 ineq1 3270 . 2  |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C
) )
31, 2ax-mp 5 1  |-  ( A  i^i  C )  =  ( B  i^i  C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1331    i^i cin 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077
This theorem is referenced by:  in12  3287  inindi  3293  dfrab2  3351  dfrab3  3352  disjpr2  3587  resres  4831  imainrect  4984  ssenen  6745  minmax  11008  xrminmax  11041  setsfun  12004  setsfun0  12005  tgrest  12348
  Copyright terms: Public domain W3C validator