Theorem r19.28zv 4465

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 1949	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4461	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ≠ wne 2959  ∀wral 3080  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-ne 2960  df-ral 3081  df-dif 4050  df-nul 4282

This theorem is used by:  raaanv  4478  raltpd  4742  iinrab  5030  iindif2  5040  iinin2  5041  reusv2lem5  5368  xpiindi  5817  dfpo2  6314  fint  6780  ixpiin  8977  neips  23458  txflf  24352  isclmp  25445  diaglbN  42303  dihglbcpreN  42548  2reuimp  48389