Theorem uneq1d 3330

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 3324	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = wceq 1373   ∪ cun 3168

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174

This theorem is referenced by:  ifeq1  3578  preq1  3715  tpeq1  3724  tpeq2  3725  resasplitss  5467  fmptpr  5789  funresdfunsnss  5800  rdgisucinc  6484  oasuc  6563  omsuc  6571  funresdfunsndc  6605  fisseneq  7046  sbthlemi5  7078  exmidfodomrlemim  7325  fzpred  10212  fseq1p1m1  10236  nn0split  10278  nnsplit  10279  fzo0sn0fzo1  10372  fzosplitprm1  10385  zsupcllemstep  10394  fsum1p  11804  fprod1p  11985  setsvala  12938  setsabsd  12946  setscom  12947  prdsex  13176  prdsval  13180  plyaddlem1  15294  plymullem1  15295