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Theorem ralf0 4409
 Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) Avoid df-clel 2830, ax-8 2113. (Revised by Gino Giotto, 2-Sep-2024.)
Hypothesis
Ref Expression
ralf0.1 ¬ 𝜑
Assertion
Ref Expression
ralf0 (∀𝑥𝐴 𝜑𝐴 = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.26 3101 . . . . 5 (∀𝑥𝐴𝜑𝜑) ↔ (∀𝑥𝐴 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑))
2 pm2.24 124 . . . . . . . 8 (𝜑 → (¬ 𝜑 → ⊥))
32impcom 411 . . . . . . 7 ((¬ 𝜑𝜑) → ⊥)
43ralimi 3092 . . . . . 6 (∀𝑥𝐴𝜑𝜑) → ∀𝑥𝐴 ⊥)
5 df-ral 3075 . . . . . . 7 (∀𝑥𝐴 ⊥ ↔ ∀𝑥(𝑥𝐴 → ⊥))
6 dfnot 1557 . . . . . . . . 9 𝑥𝐴 ↔ (𝑥𝐴 → ⊥))
76bicomi 227 . . . . . . . 8 ((𝑥𝐴 → ⊥) ↔ ¬ 𝑥𝐴)
87albii 1821 . . . . . . 7 (∀𝑥(𝑥𝐴 → ⊥) ↔ ∀𝑥 ¬ 𝑥𝐴)
95, 8sylbb 222 . . . . . 6 (∀𝑥𝐴 ⊥ → ∀𝑥 ¬ 𝑥𝐴)
10 id 22 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐴)
11 falim 1555 . . . . . . . . . . 11 (⊥ → 𝑥𝐴)
1210, 11pm5.21ni 382 . . . . . . . . . 10 𝑥𝐴 → (𝑥𝐴 ↔ ⊥))
13 df-clab 2736 . . . . . . . . . . 11 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
14 sbv 2095 . . . . . . . . . . 11 ([𝑥 / 𝑦]⊥ ↔ ⊥)
1513, 14bitri 278 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
1612, 15bitr4di 292 . . . . . . . . 9 𝑥𝐴 → (𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
1716alimi 1813 . . . . . . . 8 (∀𝑥 ¬ 𝑥𝐴 → ∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
18 dfcleq 2751 . . . . . . . 8 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
1917, 18sylibr 237 . . . . . . 7 (∀𝑥 ¬ 𝑥𝐴𝐴 = {𝑦 ∣ ⊥})
20 dfnul4 4231 . . . . . . 7 ∅ = {𝑦 ∣ ⊥}
2119, 20eqtr4di 2811 . . . . . 6 (∀𝑥 ¬ 𝑥𝐴𝐴 = ∅)
224, 9, 213syl 18 . . . . 5 (∀𝑥𝐴𝜑𝜑) → 𝐴 = ∅)
231, 22sylbir 238 . . . 4 ((∀𝑥𝐴 ¬ 𝜑 ∧ ∀𝑥𝐴 𝜑) → 𝐴 = ∅)
2423ex 416 . . 3 (∀𝑥𝐴 ¬ 𝜑 → (∀𝑥𝐴 𝜑𝐴 = ∅))
25 ralf0.1 . . . 4 ¬ 𝜑
2625a1i 11 . . 3 (𝑥𝐴 → ¬ 𝜑)
2724, 26mprg 3084 . 2 (∀𝑥𝐴 𝜑𝐴 = ∅)
28 rzal 4404 . 2 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
2927, 28impbii 212 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ⊥wfal 1550  [wsb 2069   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∅c0 4227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-ral 3075  df-dif 3863  df-nul 4228 This theorem is referenced by: (None)
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