Theorem r19.28zv 4500

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 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.28zv

Step	Hyp	Ref	Expression
1		nfv 1917	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4497	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ≠ wne 2940  ∀wral 3061  ∅c0 4322

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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-ne 2941  df-ral 3062  df-dif 3951  df-nul 4323

This theorem is referenced by:  raaanv  4521  raltpd  4785  iinrab  5072  iindif2  5080  iinin2  5081  reusv2lem5  5400  xpiindi  5835  dfpo2  6295  fint  6770  ixpiin  8920  neips  22624  txflf  23517  isclmp  24620  diaglbN  40012  dihglbcpreN  40257  2reuimp  45902