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Theorem cbviun 3910
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 2306 . . . 4 𝑦 𝑧𝐵
3 cbviun.2 . . . . 5 𝑥𝐶
43nfcri 2306 . . . 4 𝑥 𝑧𝐶
5 cbviun.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2240 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrex 2693 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
87abbii 2286 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
9 df-iun 3875 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
10 df-iun 3875 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2201 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  {cab 2156  wnfc 2299  wrex 2449   ciun 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-iun 3875
This theorem is referenced by:  cbviunv  3912  funiunfvdmf  5743  mpomptsx  6176  dmmpossx  6178  fmpox  6179  fsum2dlemstep  11397  fisumcom2  11401  fsumiun  11440  fprod2dlemstep  11585  fprodcom2fi  11589  ctiunctlemf  12393  ctiunctal  12396
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