Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.28zv Structured version   Visualization version   GIF version

Theorem r19.28zv 4418
 Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
r19.28zv (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.28zv
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜑
21r19.28z 4415 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ≠ wne 3011  ∀wral 3130  ∅c0 4265 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-dif 3911  df-nul 4266 This theorem is referenced by:  raaanv  4433  raltpd  4690  iinrab  4966  iindif2  4974  iinin2  4975  reusv2lem5  5280  xpiindi  5683  fint  6539  ixpiin  8475  neips  21716  txflf  22609  isclmp  23700  dfpo2  33065  diaglbN  38313  dihglbcpreN  38558  2reuimp  43614
 Copyright terms: Public domain W3C validator