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Theorem cbviung 4930
 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 2379. See cbviun 4928 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (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 2906 . . . 4 𝑦 𝑧𝐵
3 cbviung.2 . . . . 5 𝑥𝐶
43nfcri 2906 . . . 4 𝑥 𝑧𝐶
5 cbviung.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2837 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvrex 3358 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
87abbii 2823 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
9 df-iun 4888 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
10 df-iun 4888 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2791 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {cab 2735  Ⅎwnfc 2899  ∃wrex 3071  ∪ ciun 4886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-iun 4888 This theorem is referenced by:  cbviunvg  4934
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