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Theorem uneq1 3090
Description: Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )

Proof of Theorem uneq1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2101 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21orbi1d 705 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  \/  x  e.  C
)  <->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3084 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3084 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43bitr4g 212 . 2  |-  ( A  =  B  ->  (
x  e.  ( A  u.  C )  <->  x  e.  ( B  u.  C
) ) )
65eqrdv 2038 1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 629    = wceq 1243    e. wcel 1393    u. cun 2915
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-un 2922
This theorem is referenced by:  uneq2  3091  uneq12  3092  uneq1i  3093  uneq1d  3096  prprc1  3478  uniprg  3595  unexb  4177  relresfld  4847  relcoi1  4849  rdgeq2  5959  xpiderm  6177  findcard2  6346  findcard2s  6347  bdunexb  10014  bj-unexg  10015
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