Theorem iineq12i 36153

Description: Equality theorem for indexed intersection. Inference version. General version of iineq1i 36152. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
iineq12i.1	⊢ 𝐴 = 𝐵
iineq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
iineq12i	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷

Proof of Theorem iineq12i

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iineq12i.1	. . . 4 ⊢ 𝐴 = 𝐵
2		iineq12i.2	. . . . 5 ⊢ 𝐶 = 𝐷
3	2	eleq2i 2836	. . . 4 ⊢ (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)
4	1, 3	raleqbii 3352	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2812	. 2 ⊢ {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iin 5018	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iin 5018	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐷 = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2778	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∩ ciin 5016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-iin 5018

This theorem is referenced by: (None)