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Theorem uneq1 3084
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 2098 . . . 4 (A = B → (x Ax B))
21orbi1d 704 . . 3 (A = B → ((x A x 𝐶) ↔ (x B x 𝐶)))
3 elun 3078 . . 3 (x (A𝐶) ↔ (x A x 𝐶))
4 elun 3078 . . 3 (x (B𝐶) ↔ (x B x 𝐶))
52, 3, 43bitr4g 212 . 2 (A = B → (x (A𝐶) ↔ x (B𝐶)))
65eqrdv 2035 1 (A = B → (A𝐶) = (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242   wcel 1390  cun 2909
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916
This theorem is referenced by:  uneq2  3085  uneq12  3086  uneq1i  3087  uneq1d  3090  prprc1  3469  uniprg  3586  unexb  4143  relresfld  4790  relcoi1  4792  rdgeq2  5899  xpiderm  6113  bdunexb  9305  bj-unexg  9306
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