Theorem abeq0 3525

Description: Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)

Assertion

Ref	Expression
abeq0	⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Proof of Theorem abeq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbn 2005	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2	1	albii 1518	. 2 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3		nfv 1576	. . 3 ⊢ Ⅎ𝑦 ¬ 𝜑
4	3	sb8 1904	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5		eq0 3513	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6		df-clab 2218	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
7	6	notbii 674	. . . 4 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
8	7	albii 1518	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
9	5, 8	bitri 184	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
10	2, 4, 9	3bitr4ri 213	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1395   = wceq 1397  [wsb 1810   ∈ wcel 2202  {cab 2217  ∅c0 3494

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by:  opprc  3883