Theorem ab0OLD 4276

Description: Obsolete version of ab0 4275 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2809, ax-8 2114. (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 4225	. . 3 ⊢ ∅ = {𝑥 ∣ ⊥}
2	1	eqeq2i 2749	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥})
3		dfcleq 2729	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}))
4		df-clab 2715	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5		sb6 2093	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	bitri 278	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		df-clab 2715	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ⊥} ↔ [𝑦 / 𝑥]⊥)
8		sbv 2096	. . . . . 6 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥)
9	7, 8	bitri 278	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ ⊥} ↔ ⊥)
10	6, 9	bibi12i 343	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥))
11	10	albii 1827	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ ∀𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥))
12		nbfal 1558	. . . . 5 ⊢ (¬ ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥))
13	12	bicomi 227	. . . 4 ⊢ ((∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
14	13	albii 1827	. . 3 ⊢ (∀𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ⊥) ↔ ∀𝑦 ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
15		nfna1 2155	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)
16		nfv 1922	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜑
17		pm2.27 42	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝑦 → 𝜑) → 𝜑))
18	17	spsd 2186	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
19	18	equcoms 2030	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
20		ax12v 2178	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
21	20	equcoms 2030	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
22	19, 21	impbid 215	. . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
23	22	notbid 321	. . . 4 ⊢ (𝑦 = 𝑥 → (¬ ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ¬ 𝜑))
24	15, 16, 23	cbvalv1 2342	. . 3 ⊢ (∀𝑦 ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥 ¬ 𝜑)
25	11, 14, 24	3bitri 300	. 2 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝜑)
26	2, 3, 25	3bitri 300	1 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1541   = wceq 1543  ⊥wfal 1555  [wsb 2072   ∈ wcel 2112  {cab 2714  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-dif 3856  df-nul 4224

This theorem is referenced by: (None)