Theorem r19.9rzv 4427

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 3166	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		r19.3rzv 4426	. . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3	2	con1bid 355	. 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑))
4	1, 3	bitr2id 283	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-ne 2943  df-ral 3068  df-rex 3069  df-dif 3886  df-nul 4254

This theorem is referenced by:  r19.45zv  4430  r19.44zv  4431  r19.36zv  4434  iunconst  4930  lcmgcdlem  16239  pmtrprfvalrn  19011  dvdsr02  19813  voliune  32097  dya2iocuni  32150  filnetlem4  34497  prmunb2  41818