Theorem uneq12 4110

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 4108	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
2		uneq2 4109	. 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷))
3	1, 2	sylan9eq 2816	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∪ cun 3897

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904

This theorem is used by:  uneq12i  4113  uneq12d  4116  un00  4357  opthprc  5715  dmpropg  6216  unixp  6285  fntpg  6600  fnun  6653  resasplit  6752  fvun  6975  rankprb  9865  pm54.43  10082  pwmndgplus  19141  evlseu  22392  ptuncnv  24126  sshjval  31952  bj-2upleq  37925  bj-unexg  37951  poimirlem4  38542  poimirlem9  38547  evlselvlem  43616  diophun  43783  pwssplit4  44090  clsk1indlem3  45042