Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  difeq1 GIF version

Theorem difeq1 3187
 Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem difeq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rabeq 2678 . 2 (𝐴 = 𝐵 → {𝑥𝐴 ∣ ¬ 𝑥𝐶} = {𝑥𝐵 ∣ ¬ 𝑥𝐶})
2 dfdif2 3079 . 2 (𝐴𝐶) = {𝑥𝐴 ∣ ¬ 𝑥𝐶}
3 dfdif2 3079 . 2 (𝐵𝐶) = {𝑥𝐵 ∣ ¬ 𝑥𝐶}
41, 2, 33eqtr4g 2197 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ∈ wcel 1480  {crab 2420   ∖ cdif 3068 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-rab 2425  df-dif 3073 This theorem is referenced by:  difeq12  3189  difeq1i  3190  difeq1d  3193  uneqdifeqim  3448  diffitest  6781
 Copyright terms: Public domain W3C validator