Theorem iindif2m 3715

Description: Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
iindif2m	⊢ (∃x x ∈ A → ∩ x ∈ A (B ∖ 𝐶) = (B ∖ ∪ x ∈ A 𝐶))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   𝐶(x)

Proof of Theorem iindif2m

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28mv 3308	. . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶) ↔ (y ∈ B ∧ ∀x ∈ A ¬ y ∈ 𝐶)))
2		eldif 2921	. . . . . 6 ⊢ (y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ 𝐶))
3	2	bicomi 123	. . . . 5 ⊢ ((y ∈ B ∧ ¬ y ∈ 𝐶) ↔ y ∈ (B ∖ 𝐶))
4	3	ralbii 2324	. . . 4 ⊢ (∀x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶) ↔ ∀x ∈ A y ∈ (B ∖ 𝐶))
5		ralnex 2310	. . . . . 6 ⊢ (∀x ∈ A ¬ y ∈ 𝐶 ↔ ¬ ∃x ∈ A y ∈ 𝐶)
6		eliun 3652	. . . . . 6 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶)
7	5, 6	xchbinxr 607	. . . . 5 ⊢ (∀x ∈ A ¬ y ∈ 𝐶 ↔ ¬ y ∈ ∪ x ∈ A 𝐶)
8	7	anbi2i 430	. . . 4 ⊢ ((y ∈ B ∧ ∀x ∈ A ¬ y ∈ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∪ x ∈ A 𝐶))
9	1, 4, 8	3bitr3g 211	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∪ x ∈ A 𝐶)))
10		vex 2554	. . . 4 ⊢ y ∈ V
11		eliin 3653	. . . 4 ⊢ (y ∈ V → (y ∈ ∩ x ∈ A (B ∖ 𝐶) ↔ ∀x ∈ A y ∈ (B ∖ 𝐶)))
12	10, 11	ax-mp 7	. . 3 ⊢ (y ∈ ∩ x ∈ A (B ∖ 𝐶) ↔ ∀x ∈ A y ∈ (B ∖ 𝐶))
13		eldif 2921	. . 3 ⊢ (y ∈ (B ∖ ∪ x ∈ A 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∪ x ∈ A 𝐶))
14	9, 12, 13	3bitr4g 212	. 2 ⊢ (∃x x ∈ A → (y ∈ ∩ x ∈ A (B ∖ 𝐶) ↔ y ∈ (B ∖ ∪ x ∈ A 𝐶)))
15	14	eqrdv 2035	1 ⊢ (∃x x ∈ A → ∩ x ∈ A (B ∖ 𝐶) = (B ∖ ∪ x ∈ A 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ∖ cdif 2908  ∪ ciun 3648  ∩ ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-iun 3650  df-iin 3651

This theorem is referenced by: (None)