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Theorem raln 3105
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 3093 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
2 imnang 1804 . 2 (∀𝑥(𝑥𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
31, 2bitri 267 1 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1505  wcel 2050  wral 3088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772
This theorem depends on definitions:  df-bi 199  df-an 388  df-ral 3093
This theorem is referenced by:  ralnex  3183  rabeq0  4224
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