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Theorem ralf0 4272
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 355 . . . 4 𝜑 → (¬ 𝑥𝐴 ↔ (𝑥𝐴𝜑)))
31, 2ax-mp 5 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝜑))
43albii 1904 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝜑))
5 eq0 4130 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
6 df-ral 3101 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
74, 5, 63bitr4ri 295 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wal 1635   = wceq 1637  wcel 2156  wral 3096  c0 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-v 3393  df-dif 3772  df-nul 4117
This theorem is referenced by: (None)
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