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Theorem r19.9rzv 4499
Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.9rzv (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.9rzv
StepHypRef Expression
1 dfrex2 3073 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
2 r19.3rzv 4498 . . 3 (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥𝐴 ¬ 𝜑))
32con1bid 355 . 2 (𝐴 ≠ ∅ → (¬ ∀𝑥𝐴 ¬ 𝜑𝜑))
41, 3bitr2id 283 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wne 2940  wral 3061  wrex 3070  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-ne 2941  df-ral 3062  df-rex 3071  df-dif 3951  df-nul 4323
This theorem is referenced by:  r19.45zv  4502  r19.44zv  4503  r19.36zv  4506  iunconst  5006  lcmgcdlem  16545  pmtrprfvalrn  19358  dvdsr02  20190  voliune  33296  dya2iocuni  33351  filnetlem4  35352  prmunb2  43152
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