Theorem uneq12 4163

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 4161	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uneq2 4162	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷))
3	1, 2	sylan9eq 2797	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∪ cun 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956

This theorem is referenced by:  uneq12i  4166  uneq12d  4169  un00  4445  opthprc  5749  dmpropg  6235  unixp  6302  fntpg  6626  fnun  6682  resasplit  6778  fvun  6999  rankprb  9891  pm54.43  10041  pwmndgplus  18948  evlseu  22107  ptuncnv  23815  sshjval  31369  bj-2upleq  37013  bj-unexg  37039  poimirlem4  37631  poimirlem9  37636  evlselvlem  42596  diophun  42784  pwssplit4  43101  clsk1indlem3  44056