MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  raln Structured version   Visualization version   GIF version

Theorem raln 2985
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 2912 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
2 imnang 1766 . 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 384  wal 1478  wcel 1987  wral 2907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2912
This theorem is referenced by:  ralnex  2986  rabeq0  3931
  Copyright terms: Public domain W3C validator