Theorem uneq1d 3382

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 3376	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = 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:  ifeq1  3640  preq1  3784  tpeq1  3793  tpeq2  3794  resasplitss  5564  fmptpr  5898  funresdfunsnss  5909  rdgisucinc  6646  oasuc  6727  omsuc  6735  funresdfunsndc  6769  fisseneq  7232  sbthlemi5  7268  exmidfodomrlemim  7543  fzpred  10455  fseq1p1m1  10479  nn0split  10521  nnsplit  10522  fzo0sn0fzo1  10617  fzosplitpr  10630  fzosplitprm1  10631  zsupcllemstep  10640  hashfibclem  11260  fsum1p  12163  fprod1p  12344  setsvala  13361  setsabsd  13369  setscom  13370  prdsex  14149  prdsval  14150  plyaddlem1  15771  plymullem1  15772