Theorem r19.3rz 4095

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 3964	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		biimt 349	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
3	1, 2	sylbi 207	. 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
4		df-ral 2946	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
5		r19.3rz.1	. . . 4 ⊢ Ⅎ𝑥𝜑
6	5	19.23 2118	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
7	4, 6	bitri 264	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
8	3, 7	syl6bbr 278	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∅c0 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949

This theorem is referenced by:  r19.28z  4096  r19.3rzv  4097  r19.27z  4103  2reu4a  41510