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Theorem r19.28zv 4099
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 1883 . 2 𝑥𝜑
21r19.28z 4096 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wne 2823  wral 2941  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by:  raltpd  4346  iinrab  4614  iindif2  4621  iinin2  4622  reusv2lem5  4903  xpiindi  5290  fint  6122  ixpiin  7976  neips  20965  txflf  21857  isclmp  22943  dfpo2  31771  diaglbN  36661  dihglbcpreN  36906
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