Theorem r19.28zv 4466

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 1943	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4462	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ≠ wne 2957  ∀wral 3078  ∅c0 4285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3079  df-dif 3907  df-nul 4286

This theorem is used by:  raaanv  4479  raltpd  4746  iinrab  5032  iindif2  5042  iinin2  5043  reusv2lem5  5372  xpiindi  5820  dfpo2  6297  fint  6757  ixpiin  8920  neips  23281  txflf  24174  isclmp  25267  diaglbN  41857  dihglbcpreN  42102  2reuimp  47880