Theorem uneq2 3739

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 3738	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uncom 3735	. 2 ⊢ (𝐶 ∪ 𝐴) = (𝐴 ∪ 𝐶)
3		uncom 3735	. 2 ⊢ (𝐶 ∪ 𝐵) = (𝐵 ∪ 𝐶)
4	1, 2, 3	3eqtr4g 2680	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1480   ∪ cun 3553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560

This theorem is referenced by:  uneq12  3740  uneq2i  3742  uneq2d  3745  uneqin  3854  disjssun  4008  uniprg  4416  unexb  6911  undifixp  7888  unxpdom  8111  ackbij1lem16  9001  fin23lem28  9106  ttukeylem6  9280  lcmfun  15282  ipodrsima  17086  mplsubglem  19353  mretopd  20806  iscldtop  20809  dfconn2  21132  nconnsubb  21136  comppfsc  21245  spanun  28253  locfinref  29690  isros  30012  unelros  30015  difelros  30016  rossros  30024  inelcarsg  30154  rankung  31915  paddval  34564  dochsatshp  36220  nacsfix  36755  eldioph4b  36855  eldioph4i  36856  fiuneneq  37256  isotone1  37828  fiiuncl  38719