Theorem r19.28zv 4437

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 1921	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4434	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ≠ wne 2945  ∀wral 3066  ∅c0 4262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-9 2120  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-ne 2946  df-ral 3071  df-dif 3895  df-nul 4263

This theorem is referenced by:  raaanv  4458  raltpd  4723  iinrab  5003  iindif2  5011  iinin2  5012  reusv2lem5  5329  xpiindi  5743  dfpo2  6198  fint  6651  ixpiin  8695  neips  22262  txflf  23155  isclmp  24258  diaglbN  39065  dihglbcpreN  39310  2reuimp  44575