Theorem iineq2 5012

Description: Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iineq2	⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iineq2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2830	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
2	1	ralimi 3083	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
3		ralbi 3103	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
4	2, 3	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	abbidv 2808	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶})
6		df-iin 4994	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
7		df-iin 4994	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
8	5, 6, 7	3eqtr4g 2802	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   = 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:  iineq2i  5014  iineq2d  5015  iineq2dv  5017  firest  17477  iincld  23047  elrfirn2  42707