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Theorem ab0OLD 4306
Description: Obsolete version of ab0 4305 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2818, ax-8 2114. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ab0OLD ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Proof of Theorem ab0OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfnul4 4255 . . 3 ∅ = {𝑥 ∣ ⊥}
21eqeq2i 2752 . 2 ({𝑥𝜑} = ∅ ↔ {𝑥𝜑} = {𝑥 ∣ ⊥})
3 dfcleq 2732 . 2 ({𝑥𝜑} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}))
4 df-clab 2717 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
5 sb6 2093 . . . . . 6 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
64, 5bitri 278 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝑦𝜑))
7 df-clab 2717 . . . . . 6 (𝑦 ∈ {𝑥 ∣ ⊥} ↔ [𝑦 / 𝑥]⊥)
8 sbv 2096 . . . . . 6 ([𝑦 / 𝑥]⊥ ↔ ⊥)
97, 8bitri 278 . . . . 5 (𝑦 ∈ {𝑥 ∣ ⊥} ↔ ⊥)
106, 9bibi12i 343 . . . 4 ((𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ↔ ⊥))
1110albii 1827 . . 3 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ ∀𝑦(∀𝑥(𝑥 = 𝑦𝜑) ↔ ⊥))
12 nbfal 1558 . . . . 5 (¬ ∀𝑥(𝑥 = 𝑦𝜑) ↔ (∀𝑥(𝑥 = 𝑦𝜑) ↔ ⊥))
1312bicomi 227 . . . 4 ((∀𝑥(𝑥 = 𝑦𝜑) ↔ ⊥) ↔ ¬ ∀𝑥(𝑥 = 𝑦𝜑))
1413albii 1827 . . 3 (∀𝑦(∀𝑥(𝑥 = 𝑦𝜑) ↔ ⊥) ↔ ∀𝑦 ¬ ∀𝑥(𝑥 = 𝑦𝜑))
15 nfna1 2155 . . . 4 𝑥 ¬ ∀𝑥(𝑥 = 𝑦𝜑)
16 nfv 1922 . . . 4 𝑦 ¬ 𝜑
17 pm2.27 42 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → 𝜑))
1817spsd 2186 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
1918equcoms 2028 . . . . . 6 (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
20 ax12v 2178 . . . . . . 7 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2120equcoms 2028 . . . . . 6 (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2219, 21impbid 215 . . . . 5 (𝑦 = 𝑥 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑))
2322notbid 321 . . . 4 (𝑦 = 𝑥 → (¬ ∀𝑥(𝑥 = 𝑦𝜑) ↔ ¬ 𝜑))
2415, 16, 23cbvalv1 2343 . . 3 (∀𝑦 ¬ ∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥 ¬ 𝜑)
2511, 14, 243bitri 300 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝜑)
262, 3, 253bitri 300 1 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1541   = wceq 1543  wfal 1555  [wsb 2072  wcel 2112  {cab 2716  c0 4253
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 2016  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710
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 2717  df-cleq 2731  df-dif 3886  df-nul 4254
This theorem is referenced by: (None)
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