Theorem iineq1i 36561

Description: Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)

Hypothesis

Ref	Expression
iineq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
iineq1i	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶

Proof of Theorem iineq1i

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iineq1i.1	. . . . . 6 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2856	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	imbi1i 351	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑡 ∈ 𝐶))
4	3	ralbii2 3106	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
5	4	abbii 2831	. 2 ⊢ {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
6		df-iin 4954	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iin 4954	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
8	5, 6, 7	3eqtr4i 2797	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶

Syntax hints:   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∩ ciin 4952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-iin 4954

This theorem is referenced by: (None)