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Theorem cbviinv 5046
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2375. See cbviinvg 5048 for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbviunv.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviinv 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviinv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviunv.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
21eleq2d 2825 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
32cbvralvw 3235 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
43abbii 2807 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
5 df-iin 4999 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
6 df-iin 4999 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
74, 5, 63eqtr4i 2773 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cab 2712  wral 3059   ciin 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-iin 4999
This theorem is referenced by:  meaiininc  46443  iinhoiicc  46630  smflimlem3  46729  smflimlem4  46730  smflimlem6  46732  smfsuplem2  46768  smflimsuplem1  46776  smflimsup  46784
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