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Theorem cbviin 4494
 Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbviun.1 𝑦𝐵
cbviun.2 𝑥𝐶
cbviun.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviin 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,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 2745 . . . 4 𝑦 𝑧𝐵
3 cbviun.2 . . . . 5 𝑥𝐶
43nfcri 2745 . . . 4 𝑥 𝑧𝐶
5 cbviun.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2673 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvral 3143 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
87abbii 2726 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
9 df-iin 4458 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
10 df-iin 4458 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2642 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  ∀wral 2896  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-iin 4458 This theorem is referenced by:  cbviinv  4496  elrfirn2  36277  fnlimfvre  38741  smflimlem6  39662  smflim  39663
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