Theorem uneq2 3761

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 3760	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uncom 3757	. 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶)
3		uncom 3757	. 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶)
4	1, 2, 3	3eqtr4g 2681	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1483   ∪ cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579

This theorem is referenced by:  uneq12  3762  uneq2i  3764  uneq2d  3767  uneqin  3878  disjssun  4036  uniprg  4450  unexb  6958  undifixp  7944  unxpdom  8167  ackbij1lem16  9057  fin23lem28  9162  ttukeylem6  9336  lcmfun  15358  ipodrsima  17165  mplsubglem  19434  mretopd  20896  iscldtop  20899  dfconn2  21222  nconnsubb  21226  comppfsc  21335  spanun  28404  locfinref  29908  isros  30231  unelros  30234  difelros  30235  rossros  30243  inelcarsg  30373  noextendseq  31820  rankung  32273  paddval  35084  dochsatshp  36740  nacsfix  37275  eldioph4b  37375  eldioph4i  37376  fiuneneq  37775  isotone1  38346  fiiuncl  39234