Theorem abeq0 3553

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 2012	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
2	1	albii 1523	. 2 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
3		nfv 1581	. . 3 ⊢ Ⅎ𝑦 ¬ 𝜑
4	3	sb8 1909	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑)
5		eq0 3540	. . 3 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ {𝑥 ∣ 𝜑})
6		df-clab 2225	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
7	6	notbii 678	. . . 4 ⊢ (¬ 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ¬ [𝑦 / 𝑥]𝜑)
8	7	albii 1523	. . 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 1400   = wceq 1402  [wsb 1815   ∈ wcel 2209  {cab 2224  ∅c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-nul 3521

This theorem is referenced by:  opprc  3920