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Theorem cbviun 4747
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 2935 . . . 4 𝑦 𝑧𝐵
3 cbviun.2 . . . . 5 𝑥𝐶
43nfcri 2935 . . . 4 𝑥 𝑧𝐶
5 cbviun.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2864 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrex 3351 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
87abbii 2916 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
9 df-iun 4712 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
10 df-iun 4712 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2831 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  {cab 2785  wnfc 2928  wrex 3090   ciun 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-iun 4712
This theorem is referenced by:  cbviunv  4749  disjxiun  4840  funiunfvf  6735  mpt2mptsx  7469  dmmpt2ssx  7471  fmpt2x  7472  ovmptss  7495  iunfi  8496  fsum2dlem  14840  fsumcom2  14844  fsumiun  14891  fprod2dlem  15047  fprodcom2  15051  gsumcom2  18689  fiuncmp  21536  ovolfiniun  23609  ovoliunlem3  23612  ovoliun  23613  finiunmbl  23652  volfiniun  23655  iunmbl  23661  limciun  23999  iuneqfzuzlem  40290  fsumiunss  40547  sge0iunmpt  41374  meaiunincf  41439  meaiuninc3  41441  smfliminf  41779  dmmpt2ssx2  42910
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