Theorem difrab 3451

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 2494	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		df-rab 2494	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
3	1, 2	difeq12i 3293	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
4		df-rab 2494	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5		difab 3446	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))}
6		anass 401	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7		simpr 110	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜓)
8	7	con3i 633	. . . . . . . 8 ⊢ (¬ 𝜓 → ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))
9	8	anim2i 342	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
10		pm3.2 139	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
11	10	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝜓 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
12	11	con3d 632	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (¬ (𝑥 ∈ 𝐴 ∧ 𝜓) → ¬ 𝜓))
13	12	imdistani 445	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓))
14	9, 13	impbii 126	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	6, 14	bitr3i 186	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
16	15	abbii 2322	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))}
17	5, 16	eqtr4i 2230	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
18	4, 17	eqtr4i 2230	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
19	3, 18	eqtr4i 2230	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  {crab 2489   ∖ cdif 3167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rab 2494  df-v 2775  df-dif 3172

This theorem is referenced by: (None)