Theorem iindif1 5030

Description: Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.)

Assertion

Ref	Expression
iindif1	⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iindif1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.27zv 4464	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶)))
2		eldif 3911	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
3	2	ralbii 3082	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
4		eliin 4951	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
5	4	elv 3445	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6	5	anbi1i 624	. . . 4 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
7	1, 3, 6	3bitr4g 314	. . 3 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶)))
8		eliin 4951	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶)))
9	8	elv 3445	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶))
10		eldif 3911	. . 3 ⊢ (𝑦 ∈ (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶) ↔ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶))
11	7, 9, 10	3bitr4g 314	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ 𝑦 ∈ (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶)))
12	11	eqrdv 2734	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440   ∖ cdif 3898  ∅c0 4285  ∩ ciin 4947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-v 3442  df-dif 3904  df-nul 4286  df-iin 4949

This theorem is referenced by:  subdrgint  20736