Theorem difrab 4229

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 3115	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		df-rab 3115	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
3	1, 2	difeq12i 4048	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
4		df-rab 3115	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
5		difab 4224	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))}
6		anass 472	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓)))
7		simpr 488	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜓)
8	7	con3i 157	. . . . . . . 8 ⊢ (¬ 𝜓 → ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))
9	8	anim2i 619	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
10		pm3.2 473	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
11	10	adantr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝜓 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
12	11	con3d 155	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (¬ (𝑥 ∈ 𝐴 ∧ 𝜓) → ¬ 𝜓))
13	12	imdistani 572	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓))
14	9, 13	impbii 212	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	6, 14	bitr3i 280	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓)))
16	15	abbii 2863	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))} = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝜓))}
17	5, 16	eqtr4i 2824	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ ¬ 𝜓))}
18	4, 17	eqtr4i 2824	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} = ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∖ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
19	3, 18	eqtr4i 2824	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  {crab 3110   ∖ cdif 3878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884

This theorem is referenced by:  alephsuc3  9991  shftmbl  24142  musum  25776  clwwlknclwwlkdif  27764  aciunf1  30426  poimirlem26  35083  poimirlem27  35084