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Theorem cbviunf 29710
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 2907 . . . 4 𝑦 𝑧𝐵
5 cbviunf.2 . . . . 5 𝑥𝐶
65nfcri 2907 . . . 4 𝑥 𝑧𝐶
7 cbviunf.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
87eleq2d 2836 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
91, 2, 4, 6, 8cbvrexf 3315 . . 3 (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐴 𝑧𝐶)
109abbii 2888 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
11 df-iun 4656 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
12 df-iun 4656 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∃𝑦𝐴 𝑧𝐶}
1310, 11, 123eqtr4i 2803 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {cab 2757  wnfc 2900  wrex 3062   ciun 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-iun 4656
This theorem is referenced by:  aciunf1lem  29802
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