Theorem r19.27zv 4470

Description: Restricted quantifier version of Theorem 19.27 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
r19.27zv	⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.27zv

Step	Hyp	Ref	Expression
1		nfv 1947	. 2 ⊢ Ⅎ𝑥𝜓
2	1	r19.27z 4469	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ≠ wne 2957  ∀wral 3078  ∅c0 4282

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 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-12 2215  ax-ext 2734

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 2100  df-clab 2741  df-cleq 2754  df-ne 2958  df-ral 3079  df-dif 3905  df-nul 4283

This theorem is used by:  raaanv  4478  iindif1  5039  txflf  24233  dfso3  36286  dibglbN  42026  2alsraln0  50733