Theorem uneq1d 3325

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 3319	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   = wceq 1372   ∪ cun 3163

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169

This theorem is referenced by:  ifeq1  3573  preq1  3709  tpeq1  3718  tpeq2  3719  resasplitss  5454  fmptpr  5775  funresdfunsnss  5786  rdgisucinc  6470  oasuc  6549  omsuc  6557  funresdfunsndc  6591  fisseneq  7030  sbthlemi5  7062  exmidfodomrlemim  7308  fzpred  10191  fseq1p1m1  10215  nn0split  10257  nnsplit  10258  fzo0sn0fzo1  10348  fzosplitprm1  10361  zsupcllemstep  10370  fsum1p  11671  fprod1p  11852  setsvala  12805  setsabsd  12813  setscom  12814  prdsex  13043  prdsval  13047  plyaddlem1  15161  plymullem1  15162