Theorem difrab 3530

Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Assertion

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

Proof of Theorem difrab

Step	Hyp	Ref	Expression
1		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
2		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ ψ} = {x ∣ (x ∈ A ∧ ψ)}
3	1, 2	difeq12i 3384	. 2 ⊢ ({x ∈ A ∣ φ} ∖ {x ∈ A ∣ ψ}) = ({x ∣ (x ∈ A ∧ φ)} ∖ {x ∣ (x ∈ A ∧ ψ)})
4		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ (φ ∧ ¬ ψ)} = {x ∣ (x ∈ A ∧ (φ ∧ ¬ ψ))}
5		difab 3524	. . . 4 ⊢ ({x ∣ (x ∈ A ∧ φ)} ∖ {x ∣ (x ∈ A ∧ ψ)}) = {x ∣ ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ))}
6		anass 630	. . . . . 6 ⊢ (((x ∈ A ∧ φ) ∧ ¬ ψ) ↔ (x ∈ A ∧ (φ ∧ ¬ ψ)))
7		simpr 447	. . . . . . . . 9 ⊢ ((x ∈ A ∧ ψ) → ψ)
8	7	con3i 127	. . . . . . . 8 ⊢ (¬ ψ → ¬ (x ∈ A ∧ ψ))
9	8	anim2i 552	. . . . . . 7 ⊢ (((x ∈ A ∧ φ) ∧ ¬ ψ) → ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)))
10		pm3.2 434	. . . . . . . . . 10 ⊢ (x ∈ A → (ψ → (x ∈ A ∧ ψ)))
11	10	adantr 451	. . . . . . . . 9 ⊢ ((x ∈ A ∧ φ) → (ψ → (x ∈ A ∧ ψ)))
12	11	con3d 125	. . . . . . . 8 ⊢ ((x ∈ A ∧ φ) → (¬ (x ∈ A ∧ ψ) → ¬ ψ))
13	12	imdistani 671	. . . . . . 7 ⊢ (((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)) → ((x ∈ A ∧ φ) ∧ ¬ ψ))
14	9, 13	impbii 180	. . . . . 6 ⊢ (((x ∈ A ∧ φ) ∧ ¬ ψ) ↔ ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)))
15	6, 14	bitr3i 242	. . . . 5 ⊢ ((x ∈ A ∧ (φ ∧ ¬ ψ)) ↔ ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ)))
16	15	abbii 2466	. . . 4 ⊢ {x ∣ (x ∈ A ∧ (φ ∧ ¬ ψ))} = {x ∣ ((x ∈ A ∧ φ) ∧ ¬ (x ∈ A ∧ ψ))}
17	5, 16	eqtr4i 2376	. . 3 ⊢ ({x ∣ (x ∈ A ∧ φ)} ∖ {x ∣ (x ∈ A ∧ ψ)}) = {x ∣ (x ∈ A ∧ (φ ∧ ¬ ψ))}
18	4, 17	eqtr4i 2376	. 2 ⊢ {x ∈ A ∣ (φ ∧ ¬ ψ)} = ({x ∣ (x ∈ A ∧ φ)} ∖ {x ∣ (x ∈ A ∧ ψ)})
19	3, 18	eqtr4i 2376	1 ⊢ ({x ∈ A ∣ φ} ∖ {x ∈ A ∣ ψ}) = {x ∈ A ∣ (φ ∧ ¬ ψ)}

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619   ∖ cdif 3207

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is used by: (None)