Theorem uneq12 4116

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 4114	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uneq2 4115	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷))
3	1, 2	sylan9eq 2817	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∪ cun 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909

This theorem is referenced by:  uneq12i  4119  uneq12d  4122  un00  4399  opthprc  5711  dmpropg  6202  unixp  6269  fntpg  6581  fnun  6635  resasplit  6734  fvun  6957  rankprb  9809  pm54.43  9959  pwmndgplus  18972  evlseu  22136  ptuncnv  23867  sshjval  31553  bj-2upleq  37497  bj-unexg  37523  poimirlem4  38123  poimirlem9  38128  evlselvlem  43170  diophun  43354  pwssplit4  43666  clsk1indlem3  44619