Theorem r19.28zv 4467

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 1914	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4464	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ≠ wne 2926  ∀wral 3045  ∅c0 4299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-ne 2927  df-ral 3046  df-dif 3920  df-nul 4300

This theorem is referenced by:  raaanv  4484  raltpd  4748  iinrab  5036  iindif2  5044  iinin2  5045  reusv2lem5  5360  xpiindi  5802  dfpo2  6272  fint  6742  ixpiin  8900  neips  23007  txflf  23900  isclmp  25004  diaglbN  41056  dihglbcpreN  41301  2reuimp  47120