Theorem cbviindavw2 36329

Description: Change bound variable and domain in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbviindavw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
cbviindavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbviindavw2	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		cbviindavw2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
2	1	eleq2d 2817	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
3		cbviindavw2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
4	2, 3	cbvraldva2 3314	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷))
5	4	abbidv 2797	. 2 ⊢ (𝜑 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷})
6		df-iin 4942	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iin 4942	. 2 ⊢ ∩ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4g 2791	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∩ ciin 4940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-iin 4942

This theorem is referenced by: (None)