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

Theorem exanali 1858
Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
Assertion
Ref Expression
exanali (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))

Proof of Theorem exanali
StepHypRef Expression
1 annim 403 . . 3 ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
21exbii 1847 . 2 (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ¬ (𝜑𝜓))
3 exnal 1826 . 2 (∃𝑥 ¬ (𝜑𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))
42, 3bitri 275 1 (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1537  wex 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779
This theorem is referenced by:  gencbval  3542  dfss6  3972  nss  4047  nssss  5459  brprcneu  6895  brprcneuALT  6896  marypha1lem  9474  reclem2pr  11089  dftr6  35752  brsset  35891  dfon3  35894  dffun10  35916  elfuns  35917  ecinn0  38355  ax12indn  38945  expandrexn  44315  vk15.4j  44553  vk15.4jVD  44939
  Copyright terms: Public domain W3C validator