Theorem cbviunv 4559

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Hypothesis

Ref	Expression
cbviunv.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbviunv	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviunv

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2764	. 2 ⊢ Ⅎ𝑥𝐶
3		cbviunv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbviun 4557	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1483  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-iun 4522

This theorem is referenced by:  iunxdif2  4568  otiunsndisj  4980  onfununi  7438  oelim2  7675  marypha2lem2  8342  trcl  8604  r1om  9066  fictb  9067  cfsmolem  9092  cfsmo  9093  domtriomlem  9264  domtriom  9265  pwfseq  9486  wunex2  9560  wuncval2  9569  fsuppmapnn0fiubex  12792  s3iunsndisj  13707  ackbijnn  14560  efgs1b  18149  ablfaclem3  18486  ptbasfi  21384  bcth3  23128  itg1climres  23481  hashunif  29562  bnj601  30990  cvmliftlem15  31280  trpredlem1  31727  trpredtr  31730  trpredmintr  31731  trpredrec  31738  neibastop2  32356  filnetlem4  32376  sstotbnd2  33573  heiborlem3  33612  heibor  33620  lcfr  36874  mapdrval  36936  corclrcl  37999  trclrelexplem  38003  dftrcl3  38012  cotrcltrcl  38017  dfrtrcl3  38025  corcltrcl  38031  cotrclrcl  38034  ssmapsn  39408  cnrefiisplem  40055  cnrefiisp  40056  meaiuninclem  40697  meaiuninc  40698  meaiininc  40701  carageniuncllem2  40736  caratheodorylem1  40740  caratheodorylem2  40741  caratheodory  40742  ovnsubadd  40786  hoidmv1le  40808  hoidmvle  40814  ovnhoilem2  40816  hspmbl  40843  ovnovollem3  40872  vonvolmbl  40875  smflimlem2  40980  smflimlem3  40981  smflimlem4  40982  smflim  40985  smflim2  41012  smflimsup  41034  otiunsndisjX  41298