Theorem rabeq0 3452

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 690	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	albii 1470	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-ral 2460	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑))
4		sbn 1952	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	albii 1470	. . 3 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
6		nfv 1528	. . . 4 ⊢ Ⅎ𝑦 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)
7	6	sb8 1856	. . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑦[𝑦 / 𝑥] ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))
8		eq0 3441	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
9		df-rab 2464	. . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
10	9	eleq2i 2244	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
11		df-clab 2164	. . . . . . 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 668	. . . . 5 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
14	13	albii 1470	. . . 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 1351   = wceq 1353  [wsb 1762   ∈ wcel 2148  {cab 2163  ∀wral 2455  {crab 2459  ∅c0 3422

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-nul 3423

This theorem is referenced by:  rabnc  3455  rabrsndc  3659  exmidsssnc  4200  ssfilem  6869  diffitest  6881  ssfirab  6927  ctssexmid  7142  exmidonfinlem  7186  iooidg  9893  icc0r  9910  fznlem  10024  ioo0  10243  ico0  10245  ioc0  10246  phiprmpw  12202  hashgcdeq  12219  unennn  12378  znnen  12379