Theorem iineq12f 38193

Description: Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
iineq12f.1	⊢ Ⅎ𝑥𝐴
iineq12f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
iineq12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Proof of Theorem iineq12f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2824	. . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
2	1	ralimi 3074	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
3		ralbi 3093	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷))
4	2, 3	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷))
5		iineq12f.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6		iineq12f.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
7	5, 6	raleqf 3339	. . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
8	4, 7	sylan9bbr 510	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷))
9	8	abbidv 2802	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷})
10		df-iin 4975	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
11		df-iin 4975	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}
12	9, 10, 11	3eqtr4g 2796	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2714  Ⅎwnfc 2884  ∀wral 3052  ∩ ciin 4973

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-iin 4975

This theorem is referenced by: (None)