Theorem uneq2 4116

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 4115	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uncom 4112	. 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶)
3		uncom 4112	. 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶)
4	1, 2, 3	3eqtr4g 2797	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908

This theorem is referenced by:  uneq12  4117  uneq2i  4119  uneq2d  4122  uneqin  4243  disjssun  4422  unexbOLD  7703  undifixp  8884  unfi  9107  unxpdom  9171  ackbij1lem16  10156  fin23lem28  10262  ttukeylem6  10436  lcmfun  16584  ipodrsima  18476  mplsubglem  21966  mretopd  23048  iscldtop  23051  dfconn2  23375  nconnsubb  23379  comppfsc  23488  noextendseq  27647  oncutlt  28272  spanun  31633  constrextdg2lem  33926  locfinref  34019  isros  34346  unelros  34349  difelros  34350  rossros  34358  inelcarsg  34489  fineqvac  35294  rankung  36382  bj-funun  37507  paddval  40174  dochsatshp  41827  nacsfix  43069  eldioph4b  43168  eldioph4i  43169  fiuneneq  43549  isotone1  44404  fiiuncl  45425