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Theorem r19.9rzv 4037
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 2990 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
2 r19.3rzv 4036 . . 3 (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥𝐴 ¬ 𝜑))
32con1bid 345 . 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 2790  wral 2907  wrex 2908  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by:  r19.45zv  4040  r19.44zv  4041  r19.36zv  4044  iunconst  4495  lcmgcdlem  15243  pmtrprfvalrn  17829  dvdsr02  18577  voliune  30073  dya2iocuni  30126  filnetlem4  32018  prmunb2  37992
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