Theorem uneq12 4085

Description: Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)

Assertion

Ref	Expression
uneq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Proof of Theorem uneq12

Step	Hyp	Ref	Expression
1		uneq1 4083	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uneq2 4084	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷))
3	1, 2	sylan9eq 2853	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∪ cun 3879

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886

This theorem is referenced by:  uneq12i  4088  uneq12d  4091  un00  4350  opthprc  5580  dmpropg  6039  unixp  6101  fntpg  6384  fnun  6434  resasplit  6522  fvun  6728  rankprb  9264  pm54.43  9414  pwmndgplus  18092  evlseu  20755  ptuncnv  22412  sshjval  29133  bj-2upleq  34448  poimirlem4  35061  poimirlem9  35066  diophun  39714  pwssplit4  40033  clsk1indlem3  40746