Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  uneq1 Structured version   GIF version

Theorem uneq1 3067
 Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq1 (A = B → (A𝐶) = (B𝐶))

Proof of Theorem uneq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2083 . . . 4 (A = B → (x Ax B))
21orbi1d 692 . . 3 (A = B → ((x A x 𝐶) ↔ (x B x 𝐶)))
3 elun 3061 . . 3 (x (A𝐶) ↔ (x A x 𝐶))
4 elun 3061 . . 3 (x (B𝐶) ↔ (x B x 𝐶))
52, 3, 43bitr4g 212 . 2 (A = B → (x (A𝐶) ↔ x (B𝐶)))
65eqrdv 2020 1 (A = B → (A𝐶) = (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 616   = wceq 1228   ∈ wcel 1374   ∪ cun 2892 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899 This theorem is referenced by:  uneq2  3068  uneq12  3069  uneq1i  3070  uneq1d  3073  prprc1  3452  uniprg  3569  unexb  4127  relresfld  4774  relcoi1  4776  rdgeq2  5880  xpiderm  6088  bdunexb  7143  bj-unexg  7144
 Copyright terms: Public domain W3C validator