Theorem uneq12 4186

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 4184	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uneq2 4185	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷))
3	1, 2	sylan9eq 2800	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∪ cun 3974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981

This theorem is referenced by:  uneq12i  4189  uneq12d  4192  un00  4468  opthprc  5764  dmpropg  6246  unixp  6313  fntpg  6638  fnun  6693  resasplit  6791  fvun  7012  rankprb  9920  pm54.43  10070  pwmndgplus  18970  evlseu  22130  ptuncnv  23836  sshjval  31382  bj-2upleq  36978  bj-unexg  37004  poimirlem4  37584  poimirlem9  37589  evlselvlem  42541  diophun  42729  pwssplit4  43046  clsk1indlem3  44005