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Theorem r19.27zv 4451
 Description: Restricted quantifier version of Theorem 19.27 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
r19.27zv (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.27zv
StepHypRef Expression
1 nfv 1911 . 2 𝑥𝜓
21r19.27z 4450 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ≠ wne 3016  ∀wral 3138  ∅c0 4291 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-11 2156  ax-12 2172  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-dif 3939  df-nul 4292 This theorem is referenced by:  raaanv  4461  iindif1  4990  txflf  22608  dfso3  32945  dibglbN  38296
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