ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneq2 GIF version

Theorem uneq2 3171
Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem uneq2
StepHypRef Expression
1 uneq1 3170 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uncom 3167 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 3167 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2157 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  cun 3019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025
This theorem is referenced by:  uneq12  3172  uneq2i  3174  uneq2d  3177  uneqin  3274  disjssun  3373  uniprg  3698  sucprc  4272  unexb  4301  unfiexmid  6735  unfidisj  6739  hashunlem  10391  bdunexb  12699  bj-unexg  12700
  Copyright terms: Public domain W3C validator