Theorem r19.3rz 4462

Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Hypothesis

Ref	Expression
r19.3rz.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
r19.3rz	⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.3rz

Step	Hyp	Ref	Expression
1		n0 4318	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		biimt 360	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
3	1, 2	sylbi 217	. 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
4		df-ral 3046	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5		r19.3rz.1	. . . 4 ⊢ Ⅎ𝑥𝜑
6	5	19.23 2212	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
7	4, 6	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
8	3, 7	bitr4di 289	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∅c0 4298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-ne 2927  df-ral 3046  df-dif 3919  df-nul 4299

This theorem is referenced by:  r19.28z  4463  r19.3rzv  4464  r19.27z  4470  2reu4lem  4487