Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.9rzv Structured version   Visualization version   GIF version

Theorem r19.9rzv 4209
 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 3134 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
2 r19.3rzv 4208 . . 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 2932  ∀wral 3050  ∃wrex 3051  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-nul 4059 This theorem is referenced by:  r19.45zv  4212  r19.44zv  4213  r19.36zv  4216  iunconst  4681  lcmgcdlem  15521  pmtrprfvalrn  18108  dvdsr02  18856  voliune  30601  dya2iocuni  30654  filnetlem4  32682  prmunb2  39012
 Copyright terms: Public domain W3C validator