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Theorem cbviunf 30314
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
cbviunf.x 𝑥𝐴
cbviunf.y 𝑦𝐴
cbviunf.1 𝑦𝐵
cbviunf.2 𝑥𝐶
cbviunf.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviunf 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbviunf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviunf.x . . . 4 𝑥𝐴
2 cbviunf.y . . . 4 𝑦𝐴
3 cbviunf.1 . . . . 5 𝑦𝐵
43nfcri 2967 . . . 4 𝑦 𝑧𝐵
5 cbviunf.2 . . . . 5 𝑥𝐶
65nfcri 2967 . . . 4 𝑥 𝑧𝐶
7 cbviunf.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
87eleq2d 2899 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
91, 2, 4, 6, 8cbvrexfw 3412 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
109abbii 2887 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
11 df-iun 4896 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
12 df-iun 4896 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
1310, 11, 123eqtr4i 2855 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  {cab 2800  wnfc 2960  wrex 3131   ciun 4894
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 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-iun 4896
This theorem is referenced by:  aciunf1lem  30415
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