Theorem abeq0 3481

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 1971	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2	1	albii 1484	. 2 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3		nfv 1542	. . 3 ⊢ Ⅎ𝑦 ¬ 𝜑
4	3	sb8 1870	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5		eq0 3469	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6		df-clab 2183	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
7	6	notbii 669	. . . 4 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
8	7	albii 1484	. . 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 1362   = wceq 1364  [wsb 1776   ∈ wcel 2167  {cab 2182  ∅c0 3450

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by:  opprc  3829