Theorem uneq1d 3234

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 3228	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = wceq 1332   ∪ cun 3074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080

This theorem is referenced by:  ifeq1  3482  preq1  3608  tpeq1  3617  tpeq2  3618  resasplitss  5310  fmptpr  5620  funresdfunsnss  5631  rdgisucinc  6290  oasuc  6368  omsuc  6376  funresdfunsndc  6410  fisseneq  6828  sbthlemi5  6857  exmidfodomrlemim  7074  fzpred  9881  fseq1p1m1  9905  nn0split  9944  nnsplit  9945  fzo0sn0fzo1  10029  fzosplitprm1  10042  fsum1p  11219  zsupcllemstep  11674  setsvala  12029  setsabsd  12037  setscom  12038