Theorem abeq0 3495

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 1981	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2	1	albii 1494	. 2 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3		nfv 1552	. . 3 ⊢ Ⅎ𝑦 ¬ 𝜑
4	3	sb8 1880	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5		eq0 3483	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6		df-clab 2193	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
7	6	notbii 670	. . . 4 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
8	7	albii 1494	. . 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 1371   = wceq 1373  [wsb 1786   ∈ wcel 2177  {cab 2192  ∅c0 3464

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-nul 3465

This theorem is referenced by:  opprc  3846