Theorem abeq0 3445

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 1945	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2	1	albii 1463	. 2 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3		nfv 1521	. . 3 ⊢ Ⅎ𝑦 ¬ 𝜑
4	3	sb8 1849	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5		eq0 3433	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6		df-clab 2157	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
7	6	notbii 663	. . . 4 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
8	7	albii 1463	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
9	5, 8	bitri 183	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
10	2, 4, 9	3bitr4ri 212	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1346   = wceq 1348  [wsb 1755   ∈ wcel 2141  {cab 2156  ∅c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  opprc  3786