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Theorem exanali 1857
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 1845 . 2 (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ∃𝑥 ¬ (𝜑𝜓))
3 exnal 1824 . 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 1535  wex 1776
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
This theorem is referenced by:  gencbval  3543  dfss6  3985  nss  4060  nssss  5466  brprcneu  6897  brprcneuALT  6898  marypha1lem  9471  reclem2pr  11086  dftr6  35731  brsset  35871  dfon3  35874  dffun10  35896  elfuns  35897  ecinn0  38335  ax12indn  38925  expandrexn  44287  vk15.4j  44526  vk15.4jVD  44912
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