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  5507  fmptpr  5835  funresdfunsnss  5846  rdgisucinc  6537  oasuc  6618  omsuc  6626  funresdfunsndc  6660  fisseneq  7107  sbthlemi5  7139  exmidfodomrlemim  7390  fzpred  10278  fseq1p1m1  10302  nn0split  10344  nnsplit  10345  fzo0sn0fzo1  10439  fzosplitprm1  10452  zsupcllemstep  10461  fsum1p  11944  fprod1p  12125  setsvala  13078  setsabsd  13086  setscom  13087  prdsex  13317  prdsval  13321  plyaddlem1  15436  plymullem1  15437