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Theorem ralf0 4030
 Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.)
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
StepHypRef Expression
1 ralf0.1 . . . 4 ¬ 𝜑
2 mtt 353 . . . 4 𝜑 → (¬ 𝑥𝐴 ↔ (𝑥𝐴𝜑)))
31, 2ax-mp 5 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝜑))
43albii 1737 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝜑))
5 eq0 3888 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
6 df-ral 2901 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
74, 5, 63bitr4ri 292 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  uvtx01vtx  26020  rusgra0edg  26482
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