Theorem cbviindavw 36506

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

Hypothesis

Ref	Expression
cbviindavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)

Assertion

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

Step	Hyp	Ref	Expression
1		cbviindavw.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
2	1	eleq2d 2827	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvraldva 3221	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑡 ∈ 𝐶))
4	3	abbidv 2807	. 2 ⊢ (𝜑 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵} = {𝑡 ∣ ∀𝑦 ∈ 𝐴 𝑡 ∈ 𝐶})
5		df-iin 4927	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐵}
6		df-iin 4927	. 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑦 ∈ 𝐴 𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2801	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∀wral 3055  ∩ ciin 4925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-iin 4927

This theorem is referenced by: (None)