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Theorem cbviin 4926
 Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2379. See cbviing 4928 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypotheses
Ref Expression
cbviun.1 𝑦𝐵
cbviun.2 𝑥𝐶
cbviun.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviin 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbviin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5 𝑦𝐵
21nfcri 2906 . . . 4 𝑦 𝑧𝐵
3 cbviun.2 . . . . 5 𝑥𝐶
43nfcri 2906 . . . 4 𝑥 𝑧𝐶
5 cbviun.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2837 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvralw 3352 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
87abbii 2823 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
9 df-iin 4886 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
10 df-iin 4886 . 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  ∀wral 3070  ∩ ciin 4884 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-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  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-iin 4886 This theorem is referenced by:  cbviinv  4930  elrfirn2  40010  fnlimfvre  42682  smflimlem6  43775  smflim  43776  smflim2  43803  smfsup  43811  smfinflem  43814  smfinf  43815  smflimsup  43825  smfliminf  43828
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