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Theorem raln 3140
Description: Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.)
Assertion
Ref Expression
raln (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))

Proof of Theorem raln
StepHypRef Expression
1 df-ral 3066 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
2 imnang 1919 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
31, 2bitri 264 1 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629  wcel 2145  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-an 383  df-ral 3066
This theorem is referenced by:  ralnex  3141  rabeq0  4104
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