Theorem cbviinvw2 36633

Description: Change bound variable and domain in an indexed intersection, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbviinvw2.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
cbviinvw2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbviinvw2	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbviinvw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviinvw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
2		cbviinvw2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
3	2	eleq2d 2855	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvralvw2 36626	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2836	. 2 ⊢ {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iin 4963	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iin 4963	. 2 ⊢ ∩ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∀𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2802	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∩ ciin 4961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-iin 4963

This theorem is referenced by: (None)