Theorem rabeq0 3521

Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Assertion

Ref	Expression
rabeq0	⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)

Proof of Theorem rabeq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imnan 694	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	albii 1516	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-ral 2513	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
4		sbn 2003	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	albii 1516	. . 3 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
6		nfv 1574	. . . 4 ⊢ Ⅎ𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)
7	6	sb8 1902	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑦[𝑦 / 𝑥] ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
8		eq0 3510	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
9		df-rab 2517	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
10	9	eleq2i 2296	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
11		df-clab 2216	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
12	10, 11	bitri 184	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
13	12	notbii 672	. . . . 5 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
14	13	albii 1516	. . . 4 ⊢ (∀𝑦 ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
15	8, 14	bitri 184	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
16	5, 7, 15	3bitr4ri 213	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
17	2, 3, 16	3bitr4ri 213	1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393   = wceq 1395  [wsb 1808   ∈ wcel 2200  {cab 2215  ∀wral 2508  {crab 2512  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  rabnc  3524  rabrsndc  3734  exmidsssnc  4287  ssfilem  7045  diffitest  7057  ssfirab  7106  ctssexmid  7325  exmidonfinlem  7379  iooidg  10113  icc0r  10130  fznlem  10245  ioo0  10487  ico0  10489  ioc0  10490  phiprmpw  12752  hashgcdeq  12770  unennn  12976  znnen  12977  fczpsrbag  14643  lgsquadlem2  15765  pw0ss  15891  umgrnloop0  15925  lfgrnloopen  15939