Theorem uneq2 4091

Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
uneq2	⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Proof of Theorem uneq2

Step	Hyp	Ref	Expression
1		uneq1 4090	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uncom 4087	. 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶)
3		uncom 4087	. 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶)
4	1, 2, 3	3eqtr4g 2803	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1539   ∪ cun 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892

This theorem is referenced by:  uneq12  4092  uneq2i  4094  uneq2d  4097  uneqin  4212  disjssun  4401  uniprgOLD  4859  unexb  7598  undifixp  8722  unfi  8955  unxpdom  9030  ackbij1lem16  9991  fin23lem28  10096  ttukeylem6  10270  lcmfun  16350  ipodrsima  18259  mplsubglem  21205  mretopd  22243  iscldtop  22246  dfconn2  22570  nconnsubb  22574  comppfsc  22683  spanun  29907  locfinref  31791  isros  32136  unelros  32139  difelros  32140  rossros  32148  inelcarsg  32278  fineqvac  33066  noextendseq  33870  rankung  34468  bj-funun  35423  paddval  37812  dochsatshp  39465  nacsfix  40534  eldioph4b  40633  eldioph4i  40634  fiuneneq  41022  isotone1  41658  fiiuncl  42613