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Theorem cbviung 5034
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2365. See cbviun 5032 for a version with more disjoint variable conditions, but not requiring ax-13 2365. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbviung.1 𝑦𝐵
cbviung.2 𝑥𝐶
cbviung.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviung 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbviung
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviung.1 . . . . 5 𝑦𝐵
21nfcri 2884 . . . 4 𝑦 𝑧𝐵
3 cbviung.2 . . . . 5 𝑥𝐶
43nfcri 2884 . . . 4 𝑥 𝑧𝐶
5 cbviung.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2813 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrex 3353 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
87abbii 2796 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
9 df-iun 4992 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
10 df-iun 4992 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2764 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {cab 2703  wnfc 2877  wrex 3064   ciun 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-iun 4992
This theorem is referenced by:  cbviunvg  5038
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