Theorem difrab 3345

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 2423	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		df-rab 2423	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
3	1, 2	difeq12i 3187	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
4		df-rab 2423	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5		difab 3340	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))}
6		anass 398	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7		simpr 109	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜓)
8	7	con3i 621	. . . . . . . 8 ⊢ (¬ 𝜓 → ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))
9	8	anim2i 339	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
10		pm3.2 138	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
11	10	adantr 274	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝜓 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
12	11	con3d 620	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (¬ (𝑥 ∈ 𝐴 ∧ 𝜓) → ¬ 𝜓))
13	12	imdistani 441	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓))
14	9, 13	impbii 125	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	6, 14	bitr3i 185	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
16	15	abbii 2253	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))}
17	5, 16	eqtr4i 2161	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
18	4, 17	eqtr4i 2161	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
19	3, 18	eqtr4i 2161	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {cab 2123  {crab 2418   ∖ cdif 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068

This theorem is referenced by: (None)