ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ineq2 GIF version

Theorem ineq2 3195
Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Assertion
Ref Expression
ineq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem ineq2
StepHypRef Expression
1 ineq1 3194 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 incom 3192 . 2 (𝐶𝐴) = (𝐴𝐶)
3 incom 3192 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2145 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cin 2998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005
This theorem is referenced by:  ineq12  3196  ineq2i  3198  ineq2d  3201  uneqin  3250  intprg  3721  fiintim  6639  uzin2  10420  inopn  11600
  Copyright terms: Public domain W3C validator