Theorem r19.3rzf 45846

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypotheses

Ref	Expression
r19.3rzf.1	⊢ Ⅎ𝑥𝜑
r19.3rzf.2	⊢ Ⅎ𝑥𝐴

Assertion

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

Proof of Theorem r19.3rzf

Step	Hyp	Ref	Expression
1		r19.3rzf.2	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	n0f 4302	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		biimt 363	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
4	2, 3	sylbi 220	. 2 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)))
5		df-ral 3078	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		r19.3rzf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
7	6	19.23 2245	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
8	5, 7	bitri 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))
9	4, 8	bitr4di 292	1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566  ∃wex 1807  Ⅎwnf 1811   ∈ wcel 2141  Ⅎwnfc 2908   ≠ wne 2956  ∀wral 3077  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-dif 3907  df-nul 4286

This theorem is referenced by:  r19.28zf  45847