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

Theorem ineq2 3132
 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 3131 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 incom 3129 . 2 (𝐶𝐴) = (𝐴𝐶)
3 incom 3129 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2097 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∩ cin 2916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924 This theorem is referenced by:  ineq12  3133  ineq2i  3135  ineq2d  3138  uneqin  3188  intprg  3648  uzin2  9586
 Copyright terms: Public domain W3C validator