Theorem uneq1i 3354

Description: Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
uneq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
uneq1i	⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)

Proof of Theorem uneq1i

Step	Hyp	Ref	Expression
1		uneq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		uneq1 3351	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)

Syntax hints:   = wceq 1395   ∪ cun 3195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  un12  3362  unundi  3365  tpcoma  3760  qdass  3763  qdassr  3764  tpidm12  3765  resasplitss  5504  fmptpr  5830  df2o3  6574  undifdc  7082  sbthlemi6  7125  exmidfodomrlemim  7375  znnen  12964  setscom  13067