Theorem r19.28zv 4501

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 1910	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4498	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ≠ wne 2937  ∀wral 3058  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-ne 2938  df-ral 3059  df-dif 3950  df-nul 4324

This theorem is referenced by:  raaanv  4522  raltpd  4786  iinrab  5072  iindif2  5080  iinin2  5081  reusv2lem5  5402  xpiindi  5838  dfpo2  6300  fint  6776  ixpiin  8943  neips  23030  txflf  23923  isclmp  25037  diaglbN  40528  dihglbcpreN  40773  2reuimp  46495