Theorem r19.3rz 4433

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 4286	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		biimt 361	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
3	1, 2	sylbi 216	. 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
4		df-ral 3071	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5		r19.3rz.1	. . . 4 ⊢ Ⅎ𝑥𝜑
6	5	19.23 2208	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
7	4, 6	bitri 274	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
8	3, 7	bitr4di 289	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540  ∃wex 1786  Ⅎwnf 1790   ∈ wcel 2110   ≠ wne 2945  ∀wral 3066  ∅c0 4262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-9 2120  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-ne 2946  df-ral 3071  df-dif 3895  df-nul 4263

This theorem is referenced by:  r19.28z  4434  r19.3rzv  4435  r19.27z  4441  2reu4lem  4462