Theorem iuneq2i 4674

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 4672	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
2		iuneq2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
3	1, 2	mprg 3075	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  ∪ ciun 4655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3731  df-ss 3738  df-iun 4657

This theorem is referenced by:  dfiunv2  4691  iunrab  4702  iunid  4710  iunin1  4720  2iunin  4723  resiun1  5558  resiun1OLD  5559  resiun2  5560  dfimafn2  6389  dfmpt  6554  funiunfv  6650  fpar  7433  onovuni  7593  uniqs  7960  marypha2lem2  8499  alephlim  9091  cfsmolem  9295  ituniiun  9447  imasdsval2  16385  lpival  19461  cmpsublem  21424  txbasval  21631  uniioombllem2  23572  uniioombllem4  23575  volsup2  23594  itg1addlem5  23688  itg1climres  23702  indval2  30417  sigaclfu2  30525  measvuni  30618  trpred0  32073  rabiun  33716  mblfinlem2  33781  voliunnfl  33787  cnambfre  33791  uniqsALTV  34445  trclrelexplem  38530  cotrclrcl  38561  hoicvr  41283  hoidmv1le  41329  hoidmvle  41335  hspmbllem2  41362  smflimlem3  41502  smflimlem4  41503  smflim  41506  dfaimafn2  41767  xpiun  42295