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Theorem cbviindavw 36242
Description: Change bound variable in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbviindavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbviindavw (𝜑 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbviindavw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbviindavw.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
21eleq2d 2826 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡𝐵𝑡𝐶))
32cbvraldva 3238 . . 3 (𝜑 → (∀𝑥𝐴 𝑡𝐵 ↔ ∀𝑦𝐴 𝑡𝐶))
43abbidv 2807 . 2 (𝜑 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵} = {𝑡 ∣ ∀𝑦𝐴 𝑡𝐶})
5 df-iin 4992 . 2 𝑥𝐴 𝐵 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐵}
6 df-iin 4992 . 2 𝑦𝐴 𝐶 = {𝑡 ∣ ∀𝑦𝐴 𝑡𝐶}
74, 5, 63eqtr4g 2801 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2713  wral 3060   ciin 4990
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 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-iin 4992
This theorem is referenced by: (None)
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