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Theorem rexanali 3100
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Proof shortened by Wolf Lammen, 27-Dec-2019.)
Assertion
Ref Expression
rexanali (∃𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥𝐴 (𝜑𝜓))

Proof of Theorem rexanali
StepHypRef Expression
1 dfrex2 3071 . 2 (∃𝑥𝐴 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
2 iman 401 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
32ralbii 3091 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 ¬ (𝜑 ∧ ¬ 𝜓))
41, 3xchbinxr 335 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:  nrexralim  3135  ceqsralbv  3657  frpoind  6365  wfiOLD  6374  frind  9788  qsqueeze  13240  ncoprmgcdne1b  16684  elcls  23097  ist1-2  23371  haust1  23376  t1sep  23394  bwth  23434  1stccnp  23486  filufint  23944  fclscf  24049  pmltpc  25499  ovolgelb  25529  itg2seq  25792  radcnvlt1  26476  pntlem3  27668  nosupbnd1lem5  27772  noinfbnd1lem5  27787  umgr2edg1  29243  umgr2edgneu  29246  archiabl  33188  ordtconnlem1  33885  limsucncmpi  36428  matunitlindflem1  37603  ftc1anclem5  37684  clsk3nimkb  44030
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