Theorem r19.28zv 4472

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 1942	. 2 ⊢ Ⅎ𝑥𝜑
2	1	r19.28z 4468	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ≠ wne 2965  ∀wral 3086  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-ne 2966  df-ral 3087  df-dif 3916  df-nul 4295

This theorem is referenced by:  raaanv  4485  raltpd  4752  iinrab  5038  iindif2  5048  iinin2  5049  reusv2lem5  5377  xpiindi  5825  dfpo2  6301  fint  6761  ixpiin  8925  neips  23253  txflf  24146  isclmp  25239  diaglbN  41779  dihglbcpreN  42024  2reuimp  47801