Theorem r19.9rzv 4457

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 3062	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
2		r19.3rzv 4455	. . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
3	2	con1bid 355	. 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑))
4	1, 3	bitr2id 284	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  ∅c0 4284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-ne 2932  df-ral 3051  df-rex 3060  df-dif 3903  df-nul 4285

This theorem is referenced by:  r19.45zv  4460  r19.44zv  4461  r19.36zv  4464  iunconst  4955  lcmgcdlem  16535  pmtrprfvalrn  19419  dvdsr02  20310  voliune  34365  dya2iocuni  34419  filnetlem4  36554  prmunb2  44589