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Theorem rabeq0 3452
Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
Assertion
Ref Expression
rabeq0 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)

Proof of Theorem rabeq0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 imnan 690 . . 3 ((𝑥𝐴 → ¬ 𝜑) ↔ ¬ (𝑥𝐴𝜑))
21albii 1470 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
3 df-ral 2460 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
4 sbn 1952 . . . 4 ([𝑦 / 𝑥] ¬ (𝑥𝐴𝜑) ↔ ¬ [𝑦 / 𝑥](𝑥𝐴𝜑))
54albii 1470 . . 3 (∀𝑦[𝑦 / 𝑥] ¬ (𝑥𝐴𝜑) ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥𝐴𝜑))
6 nfv 1528 . . . 4 𝑦 ¬ (𝑥𝐴𝜑)
76sb8 1856 . . 3 (∀𝑥 ¬ (𝑥𝐴𝜑) ↔ ∀𝑦[𝑦 / 𝑥] ¬ (𝑥𝐴𝜑))
8 eq0 3441 . . . 4 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥𝐴𝜑})
9 df-rab 2464 . . . . . . . 8 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
109eleq2i 2244 . . . . . . 7 (𝑦 ∈ {𝑥𝐴𝜑} ↔ 𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
11 df-clab 2164 . . . . . . 7 (𝑦 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ [𝑦 / 𝑥](𝑥𝐴𝜑))
1210, 11bitri 184 . . . . . 6 (𝑦 ∈ {𝑥𝐴𝜑} ↔ [𝑦 / 𝑥](𝑥𝐴𝜑))
1312notbii 668 . . . . 5 𝑦 ∈ {𝑥𝐴𝜑} ↔ ¬ [𝑦 / 𝑥](𝑥𝐴𝜑))
1413albii 1470 . . . 4 (∀𝑦 ¬ 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥𝐴𝜑))
158, 14bitri 184 . . 3 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑦 ¬ [𝑦 / 𝑥](𝑥𝐴𝜑))
165, 7, 153bitr4ri 213 . 2 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
172, 3, 163bitr4ri 213 1 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)
Colors of variables: wff set class
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  3660  exmidsssnc  4203  ssfilem  6874  diffitest  6886  ssfirab  6932  ctssexmid  7147  exmidonfinlem  7191  iooidg  9908  icc0r  9925  fznlem  10040  ioo0  10259  ico0  10261  ioc0  10262  phiprmpw  12221  hashgcdeq  12238  unennn  12397  znnen  12398
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