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Theorem r19.23v 3181
Description: Restricted quantifier version of 19.23v 1940. Version of r19.23 3254 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)
Assertion
Ref Expression
r19.23v (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.23v
StepHypRef Expression
1 con34b 316 . . 3 ((𝜑𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
21ralbii 3091 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜓 → ¬ 𝜑))
3 r19.21v 3178 . 2 (∀𝑥𝐴𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥𝐴 ¬ 𝜑))
4 dfrex2 3071 . . . 4 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
54imbi1i 349 . . 3 ((∃𝑥𝐴 𝜑𝜓) ↔ (¬ ∀𝑥𝐴 ¬ 𝜑𝜓))
6 con1b 358 . . 3 ((¬ ∀𝑥𝐴 ¬ 𝜑𝜓) ↔ (¬ 𝜓 → ∀𝑥𝐴 ¬ 𝜑))
75, 6bitr2i 276 . 2 ((¬ 𝜓 → ∀𝑥𝐴 ¬ 𝜑) ↔ (∃𝑥𝐴 𝜑𝜓))
82, 3, 73bitri 297 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  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  ax-5 1908
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:  rexlimivOLD  3183  ceqsralv  3520  ralxpxfr2d  3646  uniiunlem  4097  2reu4lem  4528  dfiin2g  5037  iunss  5050  ralxfr2d  5416  ssrel2  5798  reliun  5829  idrefALT  6134  dfpo2  6318  funimass4  6973  fnssintima  7382  ralrnmpo  7572  imaeqalov  7672  ttrclss  9758  kmlem12  10200  fimaxre3  12212  gcdcllem1  16533  vdwmc2  17013  iunocv  21717  islindf4  21876  ovolgelb  25529  dyadmax  25647  itg2leub  25784  eqscut2  27866  addsprop  28024  addsuniflem  28049  negsprop  28082  mulsprop  28171  mulsuniflem  28190  mptelee  28925  nmoubi  30801  nmopub  31937  nmfnleub  31954  sigaclcu2  34101  untuni  35689  elintfv  35746  heibor1lem  37796  ispsubsp2  39729  pmapglbx  39752  neik0pk1imk0  44037  2reuimp0  47064
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