Theorem iineq12i 36370

Description: Equality theorem for indexed intersection. Inference version. General version of iineq1i 36369. (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 2827	. . . 4 ⊢ (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)
4	1, 3	raleqbii 3313	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2802	. 2 ⊢ {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iin 4948	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iin 4948	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐷 = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2768	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  ∩ ciin 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-iin 4948

This theorem is referenced by: (None)