Theorem uneq1d 3357

Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Hypothesis

Ref	Expression
uneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
uneq1d	⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))

Proof of Theorem uneq1d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		uneq1 3351	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = 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:  ifeq1  3605  preq1  3743  tpeq1  3752  tpeq2  3753  resasplitss  5504  fmptpr  5830  funresdfunsnss  5841  rdgisucinc  6529  oasuc  6608  omsuc  6616  funresdfunsndc  6650  fisseneq  7092  sbthlemi5  7124  exmidfodomrlemim  7375  fzpred  10262  fseq1p1m1  10286  nn0split  10328  nnsplit  10329  fzo0sn0fzo1  10422  fzosplitprm1  10435  zsupcllemstep  10444  fsum1p  11924  fprod1p  12105  setsvala  13058  setsabsd  13066  setscom  13067  prdsex  13297  prdsval  13301  plyaddlem1  15415  plymullem1  15416