Theorem uneq2d 3318

Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Hypothesis

Ref	Expression
uneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
uneq2d	⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Proof of Theorem uneq2d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		uneq2 3312	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1364   ∪ cun 3155

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161

This theorem is referenced by:  ifeq2  3566  tpeq3  3711  iununir  4001  unisucg  4450  relcoi1  5202  resasplitss  5440  fvun1  5630  fmptapd  5756  fvunsng  5759  fnsnsplitss  5764  tfr1onlemaccex  6415  tfrcllemaccex  6428  rdgeq1  6438  rdgivallem  6448  rdgisuc1  6451  rdgon  6453  rdg0  6454  oav2  6530  oasuc  6531  omv2  6532  omsuc  6539  fnsnsplitdc  6572  unsnfidcex  6990  undifdc  6994  fiintim  7001  ssfirab  7006  fnfi  7011  fidcenumlemr  7030  sbthlemi5  7036  sbthlemi6  7037  pm54.43  7271  fzsuc  10163  fseq1p1m1  10188  fseq1m1p1  10189  fzosplitsnm1  10304  fzosplitsn  10328  fzosplitprm1  10329  resunimafz0  10942  zfz1isolemsplit  10949  fsumm1  11600  fprodm1  11782  ennnfonelemp1  12650  ennnfonelemhdmp1  12653  ennnfonelemkh  12656  ennnfonelemhf1o  12657  ennnfonelemnn0  12666  strsetsid  12738  setscom  12745  lspun0  14059  bj-charfundcALT  15563