Theorem iuneq2i 5022

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 5020	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
2		iuneq2i.1	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
3	1, 2	mprg 3057	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∪ ciun 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-v 3464  df-ss 3964  df-iun 5003

This theorem is referenced by:  dfiunv2  5043  iunrab  5060  iunidOLD  5069  iunin1  5080  2iunin  5084  resiun1  6009  resiun2  6010  dfimafn2  6966  dfmpt  7158  funiunfv  7263  fpar  8130  onovuni  8372  uniqs  8806  marypha2lem2  9479  alephlim  10110  cfsmolem  10313  ituniiun  10465  imasdsval2  17531  lpival  21313  pzriprnglem10  21480  pzriprnglem11  21481  cmpsublem  23394  txbasval  23601  uniioombllem2  25603  uniioombllem4  25606  volsup2  25625  itg1addlem5  25721  itg1climres  25735  indval2  33847  sigaclfu2  33954  measvuni  34047  fmla  35209  rabiun  37294  mblfinlem2  37359  voliunnfl  37365  cnambfre  37369  uniqsALTV  38027  trclrelexplem  43378  cotrclrcl  43409  dfcoll2  43926  hoicvr  46169  hoidmv1le  46215  hoidmvle  46221  hspmbllem2  46248  smflimlem3  46394  smflimlem4  46395  smflim  46398  dfaimafn2  46779  xpiun  47535