Theorem ab0OLD 4309

Description: Obsolete version of ab0 4308 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2816, ax-8 2108. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ab0OLD	⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Proof of Theorem ab0OLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfnul4 4258	. . 3 ⊢ ∅ = {𝑥 ∣ ⊥}
2	1	eqeq2i 2751	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥})
3		dfcleq 2731	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}))
4		df-clab 2716	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5		sb6 2088	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	bitri 274	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		df-clab 2716	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ⊥} ↔ [𝑦 / 𝑥]⊥)
8		sbv 2091	. . . . . 6 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥)
9	7, 8	bitri 274	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ ⊥} ↔ ⊥)
10	6, 9	bibi12i 340	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥))
11	10	albii 1822	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ ∀𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥))
12		nbfal 1554	. . . . 5 ⊢ (¬ ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥))
13	12	bicomi 223	. . . 4 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
14	13	albii 1822	. . 3 ⊢ (∀𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥) ↔ ∀𝑦 ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
15		nfna1 2149	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)
16		nfv 1917	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
17		pm2.27 42	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → 𝜑))
18	17	spsd 2180	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
19	18	equcoms 2023	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
20		ax12v 2172	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
21	20	equcoms 2023	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
22	19, 21	impbid 211	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
23	22	notbid 318	. . . 4 ⊢ (𝑦 = 𝑥 → (¬ ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ¬ 𝜑))
24	15, 16, 23	cbvalv1 2338	. . 3 ⊢ (∀𝑦 ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥 ¬ 𝜑)
25	11, 14, 24	3bitri 297	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝜑)
26	2, 3, 25	3bitri 297	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  ⊥wfal 1551  [wsb 2067   ∈ wcel 2106  {cab 2715  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)