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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 3072 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
2 r19.3rzv 4498 . . 3 (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥𝐴 ¬ 𝜑))
32con1bid 355 . 2 (𝐴 ≠ ∅ → (¬ ∀𝑥𝐴 ¬ 𝜑𝜑))
41, 3bitr2id 284 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wne 2939  wral 3060  wrex 3069  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-ne 2940  df-ral 3061  df-rex 3070  df-dif 3953  df-nul 4333
This theorem is referenced by:  r19.45zv  4502  r19.44zv  4503  r19.36zv  4506  iunconst  5000  lcmgcdlem  16644  pmtrprfvalrn  19507  dvdsr02  20373  voliune  34231  dya2iocuni  34286  filnetlem4  36383  prmunb2  44335
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