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Theorem cbviindavw2 36230
Description: Change bound variable and domain in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbviindavw2.1 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
cbviindavw2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbviindavw2 (𝜑 𝑥𝐴 𝐶 = 𝑦𝐵 𝐷)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbviindavw2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 cbviindavw2.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
21eleq2d 2823 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡𝐶𝑡𝐷))
3 cbviindavw2.2 . . . 4 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
42, 3cbvraldva2 3344 . . 3 (𝜑 → (∀𝑥𝐴 𝑡𝐶 ↔ ∀𝑦𝐵 𝑡𝐷))
54abbidv 2804 . 2 (𝜑 → {𝑡 ∣ ∀𝑥𝐴 𝑡𝐶} = {𝑡 ∣ ∀𝑦𝐵 𝑡𝐷})
6 df-iin 5001 . 2 𝑥𝐴 𝐶 = {𝑡 ∣ ∀𝑥𝐴 𝑡𝐶}
7 df-iin 5001 . 2 𝑦𝐵 𝐷 = {𝑡 ∣ ∀𝑦𝐵 𝑡𝐷}
85, 6, 73eqtr4g 2798 1 (𝜑 𝑥𝐴 𝐶 = 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1535  wcel 2104  {cab 2710  wral 3057   ciin 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-iin 5001
This theorem is referenced by: (None)
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