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

Theorem ralinexa 3099
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
Assertion
Ref Expression
ralinexa (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))

Proof of Theorem ralinexa
StepHypRef Expression
1 imnan 399 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21ralbii 3091 . 2 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
3 ralnex 3070 . 2 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
42, 3bitri 275 1 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-ral 3060  df-rex 3069
This theorem is referenced by:  soseq  8183  kmlem7  10195  kmlem13  10201  lspsncv0  21166  ntreq0  23101  lhop1lem  26067  nogt01o  27756  ltrnnid  40119
  Copyright terms: Public domain W3C validator