Theorem r19.9rzv 4464

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.9rzv	⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Proof of Theorem r19.9rzv

Step	Hyp	Ref	Expression
1		dfrex2 3093	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		r19.3rzv 4462	. . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3	2	con1bid 358	. 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑))
4	1, 3	bitr2id 287	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∅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-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-ne 2960  df-ral 3081  df-rex 3091  df-dif 4050  df-nul 4282

This theorem is used by:  r19.45zv  4467  r19.44zv  4468  r19.36zv  4471  iunconst  4964  lcmgcdlem  16807  pmtrprfvalrn  19729  dvdsr02  20629  voliune  34929  dya2iocuni  34982  filnetlem4  37346  prmunb2  45485