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Theorem r19.9rzv 4206
 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 3144 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
2 r19.3rzv 4205 . . 3 (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥𝐴 ¬ 𝜑))
32con1bid 344 . 2 (𝐴 ≠ ∅ → (¬ ∀𝑥𝐴 ¬ 𝜑𝜑))
41, 3syl5rbb 273 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  ∅c0 4063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-nul 4064 This theorem is referenced by:  r19.45zv  4209  r19.44zv  4210  r19.36zv  4213  iunconst  4663  lcmgcdlem  15527  pmtrprfvalrn  18115  dvdsr02  18864  voliune  30632  dya2iocuni  30685  filnetlem4  32713  prmunb2  39036
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