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Theorem r19.3rzf 43277
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Hypotheses
Ref Expression
r19.3rzf.1 𝑥𝜑
r19.3rzf.2 𝑥𝐴
Assertion
Ref Expression
r19.3rzf (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Proof of Theorem r19.3rzf
StepHypRef Expression
1 r19.3rzf.2 . . . 4 𝑥𝐴
21n0f 4300 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
3 biimt 360 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
42, 3sylbi 216 . 2 (𝐴 ≠ ∅ → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
5 df-ral 3063 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 r19.3rzf.1 . . . 4 𝑥𝜑
7619.23 2204 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
85, 7bitri 274 . 2 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
94, 8bitr4di 288 1 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wex 1781  wnf 1785  wcel 2106  wnfc 2885  wne 2941  wral 3062  c0 4280
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-dif 3911  df-nul 4281
This theorem is referenced by:  r19.28zf  43278
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