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Theorem cbviun 4530
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
cbviun.1 𝑦𝐵
cbviun.2 𝑥𝐶
cbviun.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviun 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbviun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5 𝑦𝐵
21nfcri 2755 . . . 4 𝑦 𝑧𝐵
3 cbviun.2 . . . . 5 𝑥𝐶
43nfcri 2755 . . . 4 𝑥 𝑧𝐶
5 cbviun.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2684 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrex 3160 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
87abbii 2736 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
9 df-iun 4494 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
10 df-iun 4494 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2653 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {cab 2607  wnfc 2748  wrex 2909   ciun 4492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-iun 4494
This theorem is referenced by:  cbviunv  4532  disjxiun  4619  disjxiunOLD  4620  funiunfvf  6472  mpt2mptsx  7193  dmmpt2ssx  7195  fmpt2x  7196  ovmptss  7218  iunfi  8214  fsum2dlem  14448  fsumcom2  14452  fsumcom2OLD  14453  fsumiun  14499  fprod2dlem  14654  fprodcom2  14658  fprodcom2OLD  14659  gsumcom2  18314  fiuncmp  21147  ovolfiniun  23209  ovoliunlem3  23212  ovoliun  23213  finiunmbl  23252  volfiniun  23255  iunmbl  23261  limciun  23598  iuneqfzuzlem  39049  fsumiunss  39243  sge0iunmpt  39972  dmmpt2ssx2  41433
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