Theorem iuneq2i 4932

Description: Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Hypothesis

Ref	Expression
iuneq2i.1	⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iuneq2i	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶

Proof of Theorem iuneq2i

Step	Hyp	Ref	Expression
1		iuneq2 4930	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
2		iuneq2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
3	1, 2	mprg 3152	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∪ ciun 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3942  df-ss 3951  df-iun 4913

This theorem is referenced by:  dfiunv2  4952  iunrab  4968  iunid  4976  iunin1  4986  2iunin  4990  resiun1  5867  resiun2  5868  dfimafn2  6723  dfmpt  6900  funiunfv  7001  fpar  7805  onovuni  7973  uniqs  8351  marypha2lem2  8894  alephlim  9487  cfsmolem  9686  ituniiun  9838  imasdsval2  16783  lpival  20012  cmpsublem  22001  txbasval  22208  uniioombllem2  24178  uniioombllem4  24181  volsup2  24200  itg1addlem5  24295  itg1climres  24309  indval2  31268  sigaclfu2  31375  measvuni  31468  fmla  32623  trpred0  33070  rabiun  34859  mblfinlem2  34924  voliunnfl  34930  cnambfre  34934  uniqsALTV  35580  trclrelexplem  40049  cotrclrcl  40080  dfcoll2  40581  hoicvr  42824  hoidmv1le  42870  hoidmvle  42876  hspmbllem2  42903  smflimlem3  43043  smflimlem4  43044  smflim  43047  dfaimafn2  43359  xpiun  44027