Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralinexa GIF version

Theorem ralinexa 2351
 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 624 . . 3 ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21ralbii 2330 . 2 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑𝜓))
3 ralnex 2316 . 2 (∀𝑥𝐴 ¬ (𝜑𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
42, 3bitri 173 1 (∀𝑥𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥𝐴 (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wral 2306  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-17 1419 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator