Theorem iineq12dv 45111

Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
iineq12dv.1	⊢ (𝜑 → 𝐴 = 𝐵)
iineq12dv.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
iineq12dv	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem iineq12dv

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iineq12dv.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eleq2d 2827	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	imbi1d 341	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑡 ∈ 𝐶)))
4	3	ralbidv2 3174	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶))
5	4	abbidv 2808	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶})
6		df-iin 4994	. . 3 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iin 4994	. . 3 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∀𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2802	. 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
9		iineq12dv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
10	9	iineq2dv 5017	. 2 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐵 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
11	8, 10	eqtrd 2777	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∩ ciin 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-iin 4994

This theorem is referenced by:  smflim  46792