MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iuneq2i Structured version   Visualization version   GIF version

Theorem iuneq2i 4731
Description: Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
Hypothesis
Ref Expression
iuneq2i.1 (𝑥𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iuneq2i 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Proof of Theorem iuneq2i
StepHypRef Expression
1 iuneq2 4729 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
2 iuneq2i.1 . 2 (𝑥𝐴𝐵 = 𝐶)
31, 2mprg 3114 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156   ciun 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-v 3393  df-in 3776  df-ss 3783  df-iun 4714
This theorem is referenced by:  dfiunv2  4748  iunrab  4759  iunid  4767  iunin1  4777  2iunin  4780  resiun1  5620  resiun2  5621  dfimafn2  6463  dfmpt  6629  funiunfv  6726  fpar  7511  onovuni  7671  uniqs  8038  marypha2lem2  8577  alephlim  9169  cfsmolem  9373  ituniiun  9525  imasdsval2  16377  lpival  19450  cmpsublem  21413  txbasval  21620  uniioombllem2  23563  uniioombllem4  23566  volsup2  23585  itg1addlem5  23680  itg1climres  23694  indval2  30400  sigaclfu2  30508  measvuni  30601  trpred0  32054  rabiun  33693  mblfinlem2  33758  voliunnfl  33764  cnambfre  33768  uniqsALTV  34414  trclrelexplem  38500  cotrclrcl  38531  hoicvr  41241  hoidmv1le  41287  hoidmvle  41293  hspmbllem2  41320  smflimlem3  41460  smflimlem4  41461  smflim  41464  dfaimafn2  41752  xpiun  42331
  Copyright terms: Public domain W3C validator