Theorem r19.9rzv 4500

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 3074	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		r19.3rzv 4499	. . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3	2	con1bid 356	. 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑))
4	1, 3	bitr2id 284	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-ne 2942  df-ral 3063  df-rex 3072  df-dif 3952  df-nul 4324

This theorem is referenced by:  r19.45zv  4503  r19.44zv  4504  r19.36zv  4507  iunconst  5007  lcmgcdlem  16543  pmtrprfvalrn  19356  dvdsr02  20186  voliune  33227  dya2iocuni  33282  filnetlem4  35266  prmunb2  43070