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Theorem cbviinv 5041
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2377. See cbviinvg 5043 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 2827 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
32cbvralvw 3237 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
43abbii 2809 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
5 df-iin 4994 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
6 df-iin 4994 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
74, 5, 63eqtr4i 2775 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  wral 3061   ciin 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-iin 4994
This theorem is referenced by:  meaiininc  46502  iinhoiicc  46689  smflimlem3  46788  smflimlem4  46789  smflimlem6  46791  smfsuplem2  46827  smflimsuplem1  46835  smflimsup  46843
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