Theorem uneq1i 3379

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 3376	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)

Syntax hints:   = wceq 1402   ∪ cun 3218

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is referenced by:  un12  3387  unundi  3390  tpcoma  3801  qdass  3804  qdassr  3805  tpidm12  3806  resasplitss  5564  fmptpr  5898  df2o3  6692  undifdc  7221  sbthlemi6  7269  exmidfodomrlemim  7543  znnen  13267  setscom  13370