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Theorem ralf0 4218
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 1895 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝜑))
5 eq0 4076 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
6 df-ral 3066 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
74, 5, 63bitr4ri 293 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1629   = wceq 1631  wcel 2145  wral 3061  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by: (None)
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