Theorem uneq12 4129

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 4127	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uneq2 4128	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷))
3	1, 2	sylan9eq 2785	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∪ cun 3915

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922

This theorem is referenced by:  uneq12i  4132  uneq12d  4135  un00  4411  opthprc  5705  dmpropg  6191  unixp  6258  fntpg  6579  fnun  6635  resasplit  6733  fvun  6954  rankprb  9811  pm54.43  9961  pwmndgplus  18869  evlseu  21997  ptuncnv  23701  sshjval  31286  bj-2upleq  37007  bj-unexg  37033  poimirlem4  37625  poimirlem9  37630  evlselvlem  42581  diophun  42768  pwssplit4  43085  clsk1indlem3  44039