Theorem difrab 1695

Description: Difference of two restricted class abstractions.

Assertion

Ref	Expression
difrab	⊢ ({x ∈ A∣φ} ∖ {x ∈ A∣ψ}) = {x ∈ A∣(φ ∧ ¬ ψ)}

Proof of Theorem difrab

Step	Hyp	Ref	Expression
1		difab 1693	. . 3 ⊢ ({x∣(x ∈ A ∧ φ)} ∖ {x∣(x ∈ A ∧ ψ)}) = {x∣((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ))}
2		anass 336	. . . . 5 ⊢ (((x ∈ A ∧ φ) ∧ ¬ ψ) ↔ (x ∈ A ∧ (φ ∧ ¬ ψ)))
3		pm3.27 260	. . . . . . . 8 ⊢ ((x ∈ A ∧ ψ) → ψ)
4	3	con3i 90	. . . . . . 7 ⊢ (¬ ψ → ¬ (x ∈ A ∧ ψ))
5	4	anim2i 270	. . . . . 6 ⊢ (((x ∈ A ∧ φ) ∧ ¬ ψ) → ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)))
6		pm3.2 232	. . . . . . . . 9 ⊢ (x ∈ A → (ψ → (x ∈ A ∧ ψ)))
7	6	adantr 306	. . . . . . . 8 ⊢ ((x ∈ A ∧ φ) → (ψ → (x ∈ A ∧ ψ)))
8	7	con3d 87	. . . . . . 7 ⊢ ((x ∈ A ∧ φ) → (¬ (x ∈ A ∧ ψ) → ¬ ψ))
9	8	imdistani 340	. . . . . 6 ⊢ (((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)) → ((x ∈ A ∧ φ) ∧ ¬ ψ))
10	5, 9	impbi 139	. . . . 5 ⊢ (((x ∈ A ∧ φ) ∧ ¬ ψ) ↔ ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)))
11	2, 10	bitr3 153	. . . 4 ⊢ ((x ∈ A ∧ (φ ∧ ¬ ψ)) ↔ ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)))
12	11	biabi 1181	. . 3 ⊢ {x∣(x ∈ A ∧ (φ ∧ ¬ ψ))} = {x∣((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ))}
13	1, 12	eqtr4 1122	. 2 ⊢ ({x∣(x ∈ A ∧ φ)} ∖ {x∣(x ∈ A ∧ ψ)}) = {x∣(x ∈ A ∧ (φ ∧ ¬ ψ))}
14		df-rab 1208	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
15		df-rab 1208	. . 3 ⊢ {x ∈ A∣ψ} = {x∣(x ∈ A ∧ ψ)}
16	14, 15	difeq12i 1586	. 2 ⊢ ({x ∈ A∣φ} ∖ {x ∈ A∣ψ}) = ({x∣(x ∈ A ∧ φ)} ∖ {x∣(x ∈ A ∧ ψ)})
17		df-rab 1208	. 2 ⊢ {x ∈ A∣(φ ∧ ¬ ψ)} = {x∣(x ∈ A ∧ (φ ∧ ¬ ψ))}
18	13, 16, 17	3eqtr4 1126	1 ⊢ ({x ∈ A∣φ} ∖ {x ∈ A∣ψ}) = {x ∈ A∣(φ ∧ ¬ ψ)}

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204   ∖ cdif 1484

This theorem is referenced by:  alephsuc3 4955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-dif 1489  df-in 1491

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